+------------------------+ | Highly Regular Numbers | +------------------------+ Written by Michael Thomas De Vlieger, St. Louis, Missouri, USA Originally 20140702, fully rewritten 20170111, research finalized 201701241400, Last updated 201706191745: See update history. a(n) is the sequence generated by a necessary-but-insufficient condition for a term's inclusion in A244052. This document contains raw values of A010846(a(n)) cross-referenced between A060735, A244052, and A244053. +---------------------------------------------------------------+ | Regular numbers, counting regulars, and features in the data. | +---------------------------------------------------------------+ The nature of regular numbers and the regular counting function r(n)=OEIS A010846(n): 1. Consider two nonzero positive integers m and n; m is said to be "regular to" or "a regular of" n if and only if m | n^e for some e>=0. Regarding a(n) and A244052(n), we are concerned only with regular 1<=m<=n, i.e., regulars in the range of n. 2. Let the smallest exponent rho of n, n^rho, be the least exponent of n such that m | n^rho. This is the "richness" of regular m of n. If m | n^e, then m | n^(e+1) thus by induction, it is clear that we can find some "sure fire" power e that can serve as a congruency test for regular m | n^e, provided e>=rho. 3. It is plain from the definition in 1 that some regular m | n^1, i.e., that regular m=d | n. Thus divisors d | n are a special case of regular m of n. We can say that if d | n, then d is regular to n. Richness rho=0 for d=1, since 1 | n^0; for d>1, d | n^1 and thus rho=1 for d>1. 4. Another way of looking at regular m of n: m>1 is regular to n if and only if all prime divisors p of m also divide n. Using this definition, we would need to define the empty product m=1 regular by definition, since 1 has no prime divisors to divide n. 5. Using the definition in 4, divisors d | n are regular since all prime divisors p | d also divide n. Therefore the set of divisors d | n is a subset of the set of regulars m of n, and tau(n)<=r(n), i.e., A000005(n)<=A010846(n). In fact, if n=p prime, then tau(n)=r(n). If n = p^e (a perfect prime power with e>1), then tau(n)=r(n). Generally, if omega(n)=1, then tau(n)=r(n). Finally, if n=4, tau(n)=r(n), since 4 is the smallest composite, and divides itself. If n>4 is composite with omega(n)>1, then tau(n)4 there is at least one regular m that does not divide n, i.e., one nondivisor regular or "semidivisor" of n. n=4 cannot have semidivisors since 4 is the smallest composite. The smallest instance of a semidivisor is m=4 of n=6. 11.For the following n, r(n) = tau(n): n=1, n=p, n=p^e with e>=2, n=4. For composite n, we need to use a computation method based on (3) above, 3e being most efficient. +---------------------+ | DEFINITION OF a(n). | +---------------------+ Conditions necessary but not sufficient to generate A244052(n): Let integer 0 <= n and p_n be the n-th prime with p_0 = 1. Let p_n# = A002110(n), i.e., the product of the smallest n primes. Consider squarefree integers p_n# <= m < 2*p_n# such that omega(m) = n. Also consider multiples k*m such that 1 <= k < p_(n+1) and k*p_n <= k*m <= (k+1)*p_n. All such m > k*p_n# have r(m) > r(k*p_n), but not all are in A244052. Generally, the terms of a(n) appear in the columns A244052(n) and A054841(A244052(n)) were generated by constructing the terms of A002110/A060735 and the turbulent candidates p_T# < m < 2 p_t# such that omega(m)=T, and their "echoes". The algorithm takes advantage of the concepts of distension and depth and the relationship of cardinal distension and depth with A020900, as well as the "multiplicity notation" of A054841. Thus, a(n) can be rather efficiently constructed and all that remains is to validate the turbulent candidates and their echoes. There are other ways to examine the prime decomposition of A244052(n) besides "multiplicity notation" A054841. We can use "pi-code" A287352, which gives pi(p) of the smallest prime divisor p of A244052(n), then the first differences of the remaining primes, sorted by magnitude. Example: we can write decimal 42 as "1101" in multiplicity notation, but also "112" in pi-code. This equates to writing pi(1) = 2 for the first prime divisor of 42, then writing pi(1+1) = pi(2) = 3, finally writing pi(2+2) = pi(4) = 7. Squarefree numbers n have a zero-less pi-code. From pi-code or multiplicity notation we can derive "zerocode". This code gives the run lengths of zeros in multiplicity notation, i.e., the length of runs of prime totatives q < gpf(n). This equates to A287352(n) - 1, i.e., subtract 1 from each "digit" of pi-code. Thus for 42 = pi-code "112", we would write "001". We ignore any zeros in pi-code during conversion to zerocode, as multiplicity does not register in zerocode. Zerocode is a convenient succinct handle or label for turbulent families. For instance, dropping leading zeros from zerocode of 10, 42, 330, 2730, etc. gives us the zerocode "1". All of these numbers (10, 42, ...) have multiplicity notation that ends in "-01", that is, they have depth j=1, distension i=1, i.e., the product p_(n-1)# * p_(n+1). a(n) has been calculated to 11793 terms, i.e., all candidates for A244052 within the smallest 32 tiers, and is forthcoming as a proposed sequence in the OEIS. +----------------------------+ | Types of terms in A244052. | +----------------------------+ There are roughly three major kinds of numbers m in A244052: 1. The number m is a primorial p_T# (the product of the first T primes, i.e., in A002110). The primorials thus arrange the sequence into "tiers". All numbers m in a tier, i.e., p_T# <= m < p_(T+1)# must have omega(m) = T. The appearance of all primorials in A244052 is supported by the Mertens function version of r(n) (see Tables 6 & 7 in the Appendix) and the nature of the geometry of the table of regular 1<=m<=n as a product of distinct prime factor power tensors bounded by n (see OEIS A275280). This product is an infinite T-rank tensor I bounded by n to create a finite T-rank tensor R, and is similar to a T-dimensional simplex. See the "Tier Jump Ratio" section for data regarding the "jump" in r(n) between r(p_T#) and r((p_T - 1) p_(T-1)#), the latter the largest term of tier T-1 in A244052. 2. The number m is an integer multiple k*p_T# with 1 <= k < p_(T+1), i.e., in A060735. 3. "Turbulent" numbers m occur within a "level" k*p_T# < k*m < (k+1)*p_T# such that omega(m) = T. This is a necessary but insufficient condition. These "turbulent" terms are merely "turbulent candidates" that must be tested via the regular counting function r(m) = A010846(m) to obtain A244053(n) and compare it such that it sets a record for r(n). +-----------------------------------------+ | Major features of the tiers of A244052. | +-----------------------------------------+ There are several major features of each tier T: 1. The tier T begins with primorial p_T#. 2. All integer multiples k*p_T# with 1 <= k < p_(T+1) begin new "levels" k within each tier T. 3. There is "primary turbulence" in all tiers T>0. These turbulent terms m "echo" in level k in the form k*m and thus constitute "secondary turbulence" or "echo turbulence," also called "reverb". 4. Turbulence has the following qualities: a. Essentially, a turbulent term or candidate has gpf(m)>p_T and at least one prime totative q<=p_T such that gcd(m,p_T#) < p_T#. b. "distension" i, meaning the difference in the indexes of the greatest prime factor gpf(m) and gpf(p_T#) = p_T. This is easier to see if m is written in "multiplicity notation", that is, as A054841(m). This notation is a sort of abbreviation of highly composite numbers such that the prime is conveyed by a reverse positional notation, while only the multiplicity is written, e.g., m = 120 = 2^3 * 3^1 * 5^1 = 311; m = 630 = 2^1 * 3^2 * 5^1 * 7^1 = 1211. In this notation, primorials p_T# appear as repunits (repeated series of 1s) of length T. c. "depth" j. Let e be the index of the least prime totative (lpt) of m and let T be the index of the gpf(p_T#) = p_T. Then j = T - i. This appears in notation as the position of the first zero, e.g., the depth j = 1 of m = 10, since the second prime, 3, is coprime to 10, but the index of the gpf(10) = 3; 3 - 2 = 1, thus j = 1. Each tier has a maximum or "cardinal" depth and distension that can be calculated using A020900(T). 5. We can regard each tier of the sequences as having these structural features: Example: tier T=5: Primorial k m A054841(m) r(m) p_5# 1 2310 11111 283 <-- Head = primorial p_T# . 2730 111101 295 \ . 3570 1111001 313 |- Primary turbulence . 3990 11110001 322 | terms p_T# < m < 2p_T# 4290 * 111011 315 / with omega(m)=T 2 4620 21111 382 \ 5460 211101 395 | Cardinal distension i=3 3 6930 12111 452 | Cardinal depth j=1 8190 121101 463 | 4 9240 31111 505 |- Reverb/Chatter 10920 311101 519 | Secondary turbulence (minus terms 5 11550 11211 551 | k*p_T#, i.e., 4620, 6930, etc.) 13650 112101 567 / 6 13860 22111 593 \ 7 16170 11121 629 | 8 18480 41111 660 | 9 20790 13111 691 |- Coda/Tail -- all m of the form . 10 23100 21211 717 | k*p_T with 1 < k < p_(T+1) . 11 25410 11112 743 | v 12 27720 32111 766 / a. A "head" term that is a primorial p_T#, b. "(Primary) turbulence," wherein each term has at least one totative q <= p_T. Turbulence exhibits "distension" (the gpf(m) > p_T and there are at least 1 totative primes q < gpf(m). There is a "maximum" distension called "cardinal distension". In the above example, the cardinal distension i=3. It is plain to see in multiplicity notation (column A054841(m)); the furthest 1 to the right is 3 places from the last 1 in the multiplicity notation of p_T# with T=5 (i.e., 2310): * p_5# 1 2310 11111 283 . 2730 111101 295 . 3570 1111001 313 . 3990 11110001 322 Cardinal Distension: gpf of tier T. ---* <-- i=3; last "1" shifted three places right. Generally cardinal depth is easily computed by the following: define f(x) := pi(2 p_x) – x (pi(x)=OEIS A000720(x)) distension(T,j) = A020900(T-j+1) - (T-j+1) = f(T-j+1) Turbulence has "depth" (the least prime totative or lpt(m) < p_T). There is a "maximum" depth called "cardinal depth". In the above example, the cardinal depth is j=2. Multiplicity notation makes it easy to see the furthest 0 to the left is 1 place from the end of the "word" "11111" that is the multiplicity notation of 2310: * p_5# 1 2310 11111 283 . 2730 111101 295 . 3570 1111001 313 . 3990 11110001 322 Cardinal Depth: least prime totative of tier T. *- <-- j=1; 0 shifted one place left. If 4290 were to have qualified, cardinal 4290 * 111011 315 depth would have been *-- <-- j=2; 0 shifted two places left. Cardinal distension can be generated from the following program: jDepth[n_] := -1 + SelectFirst[Range@ n, Function[j, # Prime[n + 1]/Prime[n - j + 1] > 2 #] &[ Times @@ Prime@ Range@ n]] /. j_ /; MissingQ@ j -> 1 The notions of cardinal depth (NextPrime(gcd(t)) where t are all the terms in tier T) and distension (gpf(t)) are essential in prevalidating turbulent candidates: without it, one would be left having to test all p_T# < m < 2p_T#. See the tables in the Appendix for statistics on depth and distension in a(n), as well as confirmed volume of candidates of given depth and distension in tiers 1<=T<=14. c. Secondary turbulence, echo, chatter, reverb, etc., wherein terms of the form k*p_T# with 1 < k < p_(T+1) appear with at least one term that is k*t, where t is a term in the primary turbulence of tier T. d. The "tail" or "coda" of tier T, where all terms are of the form k*p_T# with 1 < k < p_(T+1). +------------------------------+ | The "Tier Jump Ratio." (TJR) | +------------------------------+ The ratio r( (p_(T+1)) / r( (p_(T+1)-1) p_T#) shows the "improvement" of the regular counting function r(n)=A010846(n) of tier T, specifically that of the primorial p_(T+1)# over the greatest term in tier T. The TJR continues the question, "Do primorials appear in A244052?" by adding, "How much more efficient is tier T+1 than T in producing regulars?" The TJR = 2 at T = 1, then falls to a low at T = 5, rising and falling unsteadily according to the spacing of primes, presumably. It continues to hold just over 3/2 for T = 14. Here is a table of values for each tier T (here T pertains to the primorial in the numerator): T r(p_T#) / r((p_t-1)*p_(T-1)#) 1 2 / 1 2 2 5 / 3 1.66667 3 18 / 11 1.63636 4 68 / 44 1.54545 5 283 / 192 1.47396 6 1161 / 766 1.51567 7 4843 / 3223 1.50264 8 19985 / 13037 1.53294 9 83074 / 54226 1.532 10 349670 / 230293 1.51837 11 1456458 / 942700 1.54499 12 6107257 / 3968011 1.53912 13 25547835 / 16485222 1.54974 14 106115655 / 67583798 1.57013 15 440396221 / 279200197 1.57735 It seems to follow that the ratio approaches a limit since the primes p_(T-1) and p_T tend toward similarity in magnitude on average as T increases. Some candidates for the limit, if it can be related to existing constants: 1. gamma + 1 = 1.5772156649... (gamma=Euler-Mascheroni constant). 2. pi/2 = 1.5707963267... The first candidate is interesting for the following reasons: 1. The relation between gamma and the rate of growth of the divisor counting function tau(n) = A000005(n), 2. Divisors are a special case of regular numbers 1 <= m <= n, 3. The similarity between the tensors of tau(n) and r(n) as shown by A275055 and A275280, respectively. The determination of this convergence, if there is a convergence, is beyond the scope of this work. There are smaller jumps of r(n) for each level k that have to do with increasing the bound on the infinite T-rank tensor that is the product of the prime power ranges of the distinct prime divisors p of A007947(n) (i.e., the squarefree root of n). This is perhaps the easiest "jump" effect to understand. The jumps have a logarithmic relationship to r(A007947(n)). There are even smaller jumps between turbulent candidates of the same depth that have to do with both a different limit between them and a different configuration of distinct prime divisors. The "Phantom Jump" Ratio (PJR). ------------------------------- It seems reasonable to regard TJR as a sort of gauge of "efficiency" of primorial p_T# over p_(T-1)# in generating regulars, or the higher "density" of the infinite regular tensor I_p_T over I_p_(T-1), of course taken at very nearly the same limit. Can we iron out the "very nearly the same limit" by somehow assessing what the regular function "would be" if we could measure regulars of p_T# without its greatest prime factor p_T? We can use a modified version of r(n), r'(n), to measure the number of regulars of n = p_T# attributable to all but the greatest prime factor p_T. Recognize that numbers attributable to p_T includes the products of p_T and powers p_T^e with 1<=e<=Log_p_T p_T# and any other factor of p_T#. A similar table can be produced that compares r(p_T#) to the regular function of the same number, leaving out regulars attributable to p_T. We can use the constructor approach using all prime divisors p but the largest to calculate this modified r'(n). Using the Mertens Function thus, r''(n) = r(p_T#) + mu(p_T#) floor(p_T#/p_T) = r(p_T#) - p_(T-1)# does not calculate r'(n), since all would-be totatives t attributable to p_T are not accounted for this way, only p_T itself, though it happens to furnish the first three terms below. The function r''(n) is negative for n >= 6. T r(p_T#) / r((p_t-1)*p_(T-1)#) 1 2 / 1 2 2 5 / 3 1.66667 3 18 / 12 1.5 4 68 / 46 1.47826 5 283 / 199 1.42211 6 1161 / 788 1.47335 7 4843 / 3287 1.47338 8 19985 / 13257 1.50751 9 83074 / 54963 1.51145 10 349670 / 232635 1.50308 11 1456458 / 951295 1.53103 12 6107257 / 3997121 1.52791 13 25547835 / 16590536 1.53990 14 106115655 / 67983167 1.56091 Perhaps this is more representative of the relative "density" of I_p_T over I_p_(T-1), since the difference in magnitude of p_T# and the largest term in tier T-1 does not come into play.This series seems to have a minimum at T=5 and maximum at T=1, and seems to be rising unsteadily as T increases over 5. Despite its abstract quality, maybe PJR is a better gauge than TJR of the density of regulars for primorial p_T#. See Tables 6, 7, and 8 for similar studies of partial counts of regulars of p_T# leaving out prime p_k with 1<=k<=n. Table 9 attempts to unify consideration of ignoring one prime divisor p of p_n# and ignoring p_n but considering regulars attributable to one prime q > p_n. See the coda of this document for notes and acknowledgements about computation, remarks on interesting numbers in this sequence, tables, code and open questions. A paper is forthcoming, relating proofs and conjectures regarding this sequence and its relationship with OEIS sequences A244052, A244053, A010846, A002110, A060735, A020900, A051037, A275280, and methods of computation. The paper does the "heavy lifting" regarding theorems, corollaries, and conjectures associated with this work. +-------+ | DATA. | +-------+ a(n) = position in this sequence. b(n) = position in A244052/A244053 c(n) = position in A060735 * represents a term in a(n) disqualified by r(a(n)) and thus not in A244052. (*) The "Zerocode" column gives A287352(A244052(n))-1 with all leading zeros removed. Another way to regard "zerocode" is that it is a concatenation of run lengths of zeros in A054841(A244052(n)). It is a succinct label for a given turbulent family. Position in Zerocode (*) a(n) b(n) c(n) p_T# k A244052(n) A054841(A244052(n)) A244053(n) 1 1 1 p_0# 1 1 0 1 2 2 2 p_1# 1 2 1 2 3 3 * 01 <--X 1 2 4 3 3 2 4 2 <--A 3 5 4 4 p_2# 1 6 11 5 6 5 10 101 <--B 1 6 7 6 5 2 12 21 <--C 8 8 7 6 3 18 12 10 9 8 7 4 24 31 <--D 11 10 9 8 p_3# 1 30 111 <--E 18 11 10 42 1101 1 19 12 11 9 2 60 211 26 13 12 84 2101 <--F 1 28 14 13 10 3 90 121 32 15 14 11 4 120 311 36 16 15 12 5 150 112 41 17 16 13 6 180 221 44 18 17 14 p_4# 1 210 1111 68 19 18 330 11101 1 77 20 19 390 111001 2 80 21 20 15 2 420 2111 96 22 21 16 3 630 1211 115 23 22 17 4 840 3111 131 24 23 18 5 1050 1121 145 25 24 19 6 1260 2211 156 26 25 20 7 1470 1112 166 27 26 21 8 1680 4111 174 28 27 22 9 1890 1311 183 29 28 23 10 2100 2121 192 30 29 24 p_5# 1 2310 11111 <--G 283 31 30 2730 111101 1 295 32 31 3570 1111001 2 313 33 32 3990 11110001 3 322 34 4290 * 111011 <--H 10 315 35 33 25 2 4620 21111 382 36 34 5460 211101 1 395 37 35 26 3 6930 12111 452 38 36 8190 121101 1 463 39 37 27 4 9240 31111 505 40 38 10920 311101 1 519 41 39 28 5 11550 11211 551 42 40 13650 112101 1 567 43 41 29 6 13860 22111 593 44 42 30 7 16170 11121 629 45 43 31 8 18480 41111 660 46 44 32 9 20790 13111 691 47 45 33 10 23100 21211 717 48 46 34 11 25410 11112 743 49 47 35 12 27720 32111 766 50 48 36 p_6# 1 30030 111111 1161 51 49 39270 1111101 1 1224 52 50 43890 11111001 2 1253 53 51 46410 1111011 <--I 10 1257 54 52 51870 11110101 11 1285 55 53 53130 111110001 3 1306 56 54-a 37 2 60060 211111 1526 57 55 78540 2111101 1 1597 58 56 87780 21111001 2 1631 59 57 38 3 90090 121111 1779 60 58 117810 1211101 1 1856 61 59 39 4 120120 311111 1977 62 60 40 5 150150 112111 2144 63 61 41 6 180180 221111 2294 64 62 42 7 210210 111211 2420 65 63 43 8 240240 411111 2538 66 64 44 9 270270 131111 2645 67 65 45 10 300300 212111 2743 68 66 46 11 330330 111121 2836 69 67 47 12 360360 321111 2921 70 68 48 13 390390 111112 3001 71 69 49 14 420420 211211 3080 72 70 50 15 450450 122111 3153 73 71 51 16 480480 511111 3223 74 72 52 p_7# 1 510510 1111111 4843 75 73 570570 11111101 1 4939 76 74 690690 111111001 2 5119 77 75 746130 11111011 10 5138 78 76 870870 1111110001 3 5364 79 881790 * 11110111 <--J 100 5235 80 903210 * 111110101 11 5317 81 77 930930 11111100001 4 5436 82 1009470 * 111110011 20 5412 83 78 53 2 1021020 2111111 6225 84 79 1141140 21111101 1 6337 85 80 1381380 211111001 2 6546 86 81 1492260 21111011 10 6560 87 82 54 3 1531530 1211111 7178 88 83 1711710 12111101 1 7299 89 84 55 4 2042040 3111111 7928 90 85 2282280 31111101 1 8055 91 86 56 5 2552550 1121111 8553 92 87 2852850 11211101 1 8685 93 88 57 6 3063060 2211111 9099 94 89 3423420 22111101 1 9236 95 90 58 7 3573570 1112111 9580 96 91 3993990 11121101 1 9719 97 92 59 8 4084080 4111111 10010 98 93 4564560 41111101 1 10155 99 94 60 9 4594590 1311111 10414 100 95 61 10 5105100 2121111 10777 101 96 62 11 5615610 1111211 11120 102 97 63 12 6126120 3211111 11441 103 98 64 13 6636630 1111121 11740 104 99 65 14 7147140 2112111 12027 105 100 66 15 7657650 1221111 12293 106 101 67 16 8168160 5111111 12549 107 102 68 17 8678670 1111112 12799 108 103 69 18 9189180 2311111 13037 109 104-b 70 p_8# 1 9699690 11111111 19985 110 105 11741730 111111101 1 20605 111 106 13123110 111111011 10 20929 112 107 14804790 1111111001 2 21453 113 108 15825810 11111110001 3 21713 114 109 16546530 1111110101 11 21769 115 17160990 * 111110111 100 21559 116 110 17687670 11111101001 12 22028 117 111 18888870 111111100001 4 22443 118 112 71 2 19399380 21111111 25289 119 113 23483460 211111101 1 26005 120 114 26246220 211111011 10 26370 121 115 72 3 29099070 12111111 28924 122 116 35225190 121111101 1 29701 123 117 73 4 38798760 31111111 31776 124 118 46966920 311111101 1 32594 125 119 74 5 48498450 11211111 34150 126 120 75 6 58198140 22111111 36204 127 121 76 7 67897830 11121111 38028 128 122 77 8 77597520 41111111 39660 129 123 78 9 87297210 13111111 41161 130 124 79 10 96996900 21211111 42543 131 125 80 11 106696590 11112111 43827 132 126 81 12 116396280 32111111 45029 133 127 82 13 126095970 11111211 46156 134 128 83 14 135795660 21121111 47233 135 129 84 15 145495350 12211111 48240 136 130 85 16 155195040 51111111 49202 137 131 86 17 164894730 11111121 50130 138 132 87 18 174594420 23111111 51014 139 133 88 19 184294110 11111112 51861 140 134 89 20 193993800 31211111 52680 141 135 90 21 203693490 12121111 53468 142 136 91 22 213393180 21112111 54226 143 137 92 p_9# 1 223092870 111111111 83074 144 138 281291010 1111111101 1 86054 145 139 300690390 11111111001 2 86978 146 140 340510170 1111111011 10 88168 147 141 358888530 111111110001 3 89598 148 363993630 * 11111110101 11 89097 149 380570190 * 1111110111 100 89235 150 142 397687290 1111111100001 4 91214 151 406816410 * 11111101101 101 90163 152 143 417086670 11111111000001 5 91993 153 434444010 * 111111101001 12 91713 154 144 93 2 446185740 211111111 103747 155 145 562582020 2111111101 1 107188 156 146 601380780 21111111001 2 108267 157 147 94 3 669278610 121111111 117837 158 148 843873030 1211111101 1 121572 159 149-c 95 4 892371480 311111111 128844 160 150 96 5 1115464350 112111111 137989 161 151 97 6 1338557220 221111111 145890 162 152 98 7 1561650090 111211111 152876 163 153 99 8 1784742960 411111111 159160 164 154 100 9 2007835830 131111111 164894 165 155 101 10 2230928700 212111111 170187 166 156 102 11 2454021570 111121111 175094 167 157 103 12 2677114440 321111111 179692 168 158 104 13 2900207310 111112111 184000 169 159 105 14 3123300180 211211111 188085 170 160 106 15 3346393050 122111111 191949 171 161 107 16 3569485920 511111111 195627 172 162 108 17 3792578790 111111211 199150 173 163 109 18 4015671660 231111111 202514 174 164 110 19 4238764530 111111121 205745 175 165 111 20 4461857400 312111111 208859 176 166 112 21 4684950270 121211111 211853 177 167 113 22 4908043140 211121111 214739 178 168 114 23 5131136010 111111112 217534 179 169 115 24 5354228880 421111111 220238 180 170 116 25 5577321750 113111111 222860 181 171 117 26 5800414620 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39 510227691935131170 12111211111111 268169500 646 556 311 40 523310453266801200 41211111111111 269834638 647 557 312 41 536393214598471230 11111111111121 271467914 648 558 313 42 549475975930141260 22121111111111 273070560 649 559 314 43 562558737261811290 11111111111112 274643977 650 560 315 44 575641498593481320 31112111111111 276189318 651 561 316 45 588724259925151350 13211111111111 277707756 652 562 317 46 601807021256821380 21111111211111 279200197 653 563 318-f p_15# 1 614889782588491410 111111111111111 440396221 +--------------------+ | Research Progress. | +--------------------+ Research pertaining to the regular function of all candidates a(1)-a(652) has concluded. The Loopsweep program of 23 May 2017 improved the efficiency of generating squarefree p_n# <= m <= 2p_n# that have omega(m) = n. It uses a loop to direct iteration of A287352(m) such that the tree of all such m generates with minimal unqualified m. The Permutations-based sweep program created a large volume of unqualified m and had an effective limit at around tier 40. Loopsweep has produced results up to tier 80-100. Loopsweep might not contribute to the extension of A244052, but will aid in generating ancillary sequences such as the tables in this document. +----------+ | Remarks. | +----------+ The following numbers are remarkable in aspects having to do with general number theory, in the structure of their primes, or their qualification as a term in A244052. Firstly, 1 and 2 are in the sequences a(n) and A244052 because they are the empty product and the smallest primorial != 1, respectively. The number 2 sets up the first tier, T=1. The number 1 is the only odd number in the sequence. The number 2 is the only prime in the sequence; r(2) / r(1) = 2 sets the highest value seen for the "tier jump ratio". (X) - 3 is the second prime in a(n) but r(2) = r(3), thus it does not enter A244052. It is the first turbulent term with the class -01 and depth j = 1. (A) - 4 is the smallest term of the form k*p_T# with 1=1 have "levels" organized by k*p_T#. (B) - The primorial 6 sets the second greatest-seen value for the "tier jump ratio". r(6) / r(4) = 5/3 = 1.6666... The number 6 is the first squarefree composite in the sequence. (C) - 10 is the smallest "turbulent" term, depth j=1, distension i=1. 10 is of the notation class -01, which represents the smallest possible turbulent term. The class -01 appears in all tiers T>=2. The term "turbulence" refers to the appearance of the A054841 "multiplicity notation" of m in tier T. (D) - 24 is the smallest term of A244052 that is not in A275881, the sequence of numbers that have at least n/2 regulars. This is to say that 24 has less than 24/2=12 regulars. A275881 = {1, 2, 3, 4, 6, 8, 10, 12, 18, 30}. Of these, {3, 4, 8} are not in a(n) nor A244052, since they are odd primes or perfect prime powers. (E) - 30 is the largest term in A275881, with 18 regulars, it is the largest number to have at least half of its range 1<=k<=n regular. Additionally, 30 is the largest number in A020490 = {1, 2, 3, 4, 6, 8, 10, 12, 18, 24, 30}: numbers that have at least as many divisors as totatives. The number 30 is also the largest number without composite totatives. (F) - 84 is the smallest "echo turbulent" term, appearing in the "reverb" segment of tier T=3. All tiers T>=3 have "reverb" segments that come after the "primary turbulence" and precede the "tail" wherein all terms are of the form k*p_T#. 84 is the "echo" of 42: both are of the class -01. (G) - The primorial 2310 sets the least-seen value for the "tier jump ratio", r(2310) / r(2100) = 183/192 = 1.6666... If we count the primorial 2, then r(2) / r(1) = 2 sets the highest value. (H) - 4290 is the first composite turbulent candidate disqualified by r(a(n)) - r(a(n-1)), i.e., its regular function value 315 is lower than that of the preceding candidate, 3990, with r(3990) = 322. Further, 4290 is of the class -011, the first candidate with depth j=2. From this, it might seem to follow that all terms with depth j=2 disqualify... Tier 5 thus has the first disqualification of a term of p_T# < a(n) < 2p_T# with omega(a(n)) = T. Though tier 5's 4290 disqualifies, no term of a(n) disqualifies in tier 6. Tier T=6 is conjectured to be the greatest T with no disqualifying terms of a(n). Tier T=8 has one disqualification (17160990, class -0111). (I) - 46410 in tier T=6 is the smallest qualifying candidate with depth j=2, of class -011. With r(46410)=1257, it bests the preceding term 43890 (class -001) by just 4 regulars. All tiers T>=6 have depth-2 turbulent terms, and tier T=6 is the smallest tier with qualified depth j=2 turbulence. (J) - 881790 is the smallest depth j=3 candidate; r(881790) is insufficient to qualify it. 881790 is of the class -0111; tier T=8's 17160990 and T=9's 380570190 are also of this class and depth; the former the only candidate disqualified in tier 8. These observations may lead us to conjecture that no term with j>2 can qualify... (K) - 11797675890 of tier T=10 is the smallest depth j=4 candidate, but is disqualified in tiers 10 through 14... (L) - Tier 11 is the smallest tier that has more depth j=2 candidates than j=1. (M) - 10491388397490 is the smallest depth j=3 term in A244052, it is of the class -0111 in tier 12. This tier also qualifies a second depth-3 term 13326898775190 of class -010101. Thus tier T=12 is the smallest tier with qualified depth j=3 turbulence. (N) - 22006326882540 is the smallest echo-turbulent (secondary-turbulent) disqualification. It is the "echo" of 11003163441270, disqualified in the primary turbulence ot tier T=12. This supports the conjecture that a candidate that fails to qualify in primary turbulence will not qualify in higher levels k>1. In all smaller tiers T, the smallest disqualifying candidate in primary turbulence, t, was sufficiently large such that 2t > 3*p_T# and thereby did not appear in secondary turbulence. This is the smallest instance that 2t < 3*p_T#. (O) - 568815710072610 in tier T=13 is the smallest depth j=5 candidate, but is disqualified in tier 13... Tier 13 has 44 turbulent candidates but disqualifies 23; it is the first tier to disqualify more than half of the turbulent candidates predicted by the necessary- but-insufficient condition described above. Tier 12 has 31 but disqualifies 13. We conjecture that tier 13 is the smallest T where more turbulent terms of a(n) are not such in A244052, and all larger T disqualify more than half such candidates. An exception may be one or more smaller T > 13. It is also interesting to observe the regular function, i.e., the values of c = A010846(a(n)) holding narrowly in the channel r(p_13#) = 25547835 < r(c) < 27862209. This supports the conjecture that r(p_T#) necessarily sets the floor for r(c), and that all c in tier T of a(n) have r(c) > r(p_T#). (P) - 902259402184140 in tier 13 is the echo-disqualification of the depth j=4, class -01111 candidate in level k=2. Do echo-disqualifications happen in every tier T>12? (The answer seems to be yes.) Is the apparent "heritability" of qualification/disqualification attributable to the size of numbers as the tiers increase, since r(n) is not multiplicative? (e.g., r(6)=5 but r(2*6)=8. Indeed r(451129701092070)=26608037 but r(902259402184140)=31986440). +--------+ | Notes. | +--------+ General: all figures in this document were generated using Wolfram Mathematica. Were someone able to compile r(n) it is anticipated that they could reach tiers 19-20 were they to tolerate daylong processing per term near the end of their research. The forthcoming paper has information about the cardinal depth and distension for tiers up to 36 and anticipated turbulent population. (a) - Terms 1-54 calculated via determination that the distinct prime divisors of m <= n are a subset of those of n. The process is relatively slow. A244052 an A244053 were proposed 18 June 2014. (b) - Terms 55-104 were predicted in large part by an early conjecture related to the proofs in the forthcoming paper. They were calculated 9 February 2015 by David Corneth. Along with these terms, David Corneth also calculated r(A002110(T)) for 9 <= T <= 13 (cf. OEIS A010846). (c) - Terms 105-149 were furnished by David Corneth and validated using a test related to the fact that regular m | n^m. (a more suitable test is m | n^floor(log2(n))). See OEIS A280269 and A280274). (d) - Terms 150-188 were calculated in November 2016 using the m | n^m test, The terms were validated by an algorithm generalized by those seen at A051037, now called the "constructor" approach. (e) - Terms that are k*p_T# or k*t (t being a term in primary turbulence) can be processed together. For example, if we want to determine the regulars of all terms in the range k*p_T# with integer 1<=k 4. Is there a reliable method for counting regulars that does not involve testing or construction of a tensor? Can we use calculus to compute the volume of a T-dimensional simplex-like figure with a complex curved face, then take a floor-function-like approach and arrive at a close approximation? 5. Is there a primary turbulent candidate m that is echoed as k*p_T# < k*m < (k+1)*p_T# that does not enter A244052 while k*m does (or vice versa)? This is to ask, does the inclusion/disclusion of k*m necessitate the same for (k+1)m? (My sense says no). 6. Is there a smaller "missing" of r(a(n-1) by r(a(n)) than 7? (cf. Remark H). +-----------+ | Appendix. | +-----------+ Let "distension" i = pi(gpf(t))-(T+1), where t is a turbulent term in tier T, and T+1=pi(NextPrime(p_T)). Let "depth" j = (T+1)-pi(lpt(t)), with lpf(n)= the least prime that is coprime to n. We can use distension and depth as iterators for a function that can construct the turbulent candidates of tier T, eliminating the need to search p_T#<=m1 in A244052. This is because a small lpt incurs greater damage to the total value of r(n) (cf. the Mertens function version of r(n)). Depending on how a term of depth j>1 compares in magnitude and thereby place in the sorted turbulent candidates t, particularly vis a vis terms of depth j=1, the deep candidate may be simply eclipsed by the shallower one. 4. A term of depth j>1 may appear in tier T of A244052, but not in tier T+1, mainly because of the arrangement of turbulent terms of varying depths. 5. Very deep turbulent terms seem unlikely to appear in A244052. The exact relationship of deepest turbulent term to tier is not well understood. Table 4. Primary turbulent candidate volume (right, excerpted from above) vs. confirmed primary turbulent volume in A244052. 1| 0 1| 0 2| 1 2| 1 3| 1 3| 1 T 4| 2 T 4| 2 I 5| 3 1 I 5| 3 . E 6| 3 2 E 6| 3 2 R 7| 4 3 1 R 7| 4 1 . 8| 4 3 1 8| 4 3 . 9| 5 3 2 9| 5 1 . 10| 6 7 2 1 10| 6 4 . . 11| 7 9 3 1 11| 7 6 . . 12| 9 11 8 2 12| 9 7 2 . 13| 9 16 11 6 1 13| 9 9 3 . . 14| 9 16 15 6 3 14| 9 10 2 . . 15| 9 15 14 9 3 1 15| 9? ? ? ? . . –– –– –– –– –– –– –– –– –– –– –– –– –– –– 1 2 3 4 5 6 1 2 3 4 5 6 DEPTH DEPTH The data in Table 4 supports the conjectures on depth. We await research results for tier 15, which lies beyond the scope of this study but may come at a later date. Table 5. Summary of various aggregate aspects of each tier T of a(n). The first column lists the tier T. The next column is prime(T), e.g. prime(2)=3, OEIS A000040. The cardinal distension i_T and cardinal depth j_T appear in the next two columns. The total of all j-distensions (i.e., the sum of i values for each depth j, row sums of Table 1) appears in the fifth column. The primary volume columns list the number of distinct turbulent terms t with depth j = 1 for a(n) (marked "a") and A244052 (marked "A"). Values in "Primary Volume a" are row sums of Table 3. The next columns titled "Total Volume" show the total number of turbulent terms in the tier, including reverberations across levels. The "Primorial Position" columns show the location of primorial p_T# in a(n) and A244052. The "Population" columns give the total number of terms in tier T of a(n) and A244052. Primary Total Primorial Population Sum Volume Volume Position T p(T) i j (i,j) a A a A a A a A 0| * 0 0 0 0 0 0 0 1 1 1 1 1| 2 0 0 0 0 0 0 0 2 2 2 2 2| 3 1 1 1 1 1 1 1 4 4 5 5 3| 5 1 1 1 1 1 2 2 9 9 8 8 4| 7 2 1 2 2 2 2 2 17 17 12 12 5| 11 3 2 4 4 4 8 7 29 29 20 19 6| 13 3 2 5 5 5 8 8 49 48 24 24 7| 17 4 3 7 8 5 17 14 73 72 25 22 8| 19 4 3 8 8 7 12 11 108 104 34 33 9| 23 5 3 10 10 9 13 9 142 137 41 37 10| 29 6 4 13 16 10 34 28 183 174 64 58 11| 31 7 4 16 20 13 31 24 247 232 67 60 12| 37 9 4 21 30 18 51 38 314 292 91 77 13| 41 9 5 26 43 21 83 60 405 369 125 101 14| 43 9 5 30 49 26 76 52 530 470 122 93 15| 47 9 6 34 51 72 652 563 124 16| 53 11 6 39 68 95 776 153 17| 59 13 7 46 100 162 929 222 18| 61 12 7 51 115 163 1151 229 19| 67 13 8 57 156 228 1380 298 20| 71 14 9 63 199 301 1678 373 21| 73 13 9 67 205 274 2051 352 22| 79 15 9 73 237 316 2403 398 23| 83 15 9 79 259 331 2801 419 24| 89 16 9 86 307 388 3220 484 25| 97 19 10 96 452 599 3704 699 26| 101 20 11 106 624 865 4403 967 27| 103 19 11 114 715 951 5370 1057 28| 107 19 12 122 776 1037 6427 1145 29| 109 18 13 128 694 865 7572 977 30| 113 18 12 133 575 669 8549 795 31| 127 23 13 144 792 959 9344 1089 32| 131 23 13 154 1018 1225 10433 1361 33| 137 25 14 166 1344 1653 11794 1791 34| 139 25 13 177 1498 1748 13585 1896 35| 149 27 14 191 2088 2502 15481 2652 36| 151 26 15 203 2465 2862 18133 3018 37| 157 28 15 216 2973 3418 21151 3580 38| 163 28 15 229 3690 4225 24731 4391 39| 167 28 16 242 4259 4818 29122 4990 40| 173 28 16 254 5046 5687 34112 5865 41| 179 30 17 268 6129 6943 39977 7123 42| 181 30 18 281 6378 7078 47100 7268 43| 191 32 19 295 8060 9054 54368 9246 44| 193 32 19 300 8762 9743 63614 9939 45| 197 32 20 321 8933 9881 73553 10079 46| 199 32 19 333 7766 8388 83632 8598 47| 211 35 18 349 8897 9586 92230 9808 48| 223 38 18 369 13273 14438 102038 14664 49| 227 38 19 389 18495 20252 116702 20480 50| 229 38 20 408 22259 24212 137182 24444 51| 233 39 21 427 25086 27113 161626 27351 52| 239 39 22 445 28600 30872 188977 31112 53| 241 39 23 462 28225 30151 220089 30401 54| 251 41 23 480 32798 34983 250490 35239 55| 257 42 23 499 38421 40959 285729 41221 56| 263 43 24 519 45419 48432 326950 48700 57| 269 42 25 537 54061 57760 375650 58030 58| 271 42 25 554 56105 59676 433680 59952 59| 277 42 25 571 58491 62101 493632 62381 60| 281 42 26 588 57035 60397 556013 60679 Table 6. Partial Mertens function applied to primorials p_n# achieved by leaving out prime p_k with 1<=k<=n, its powers p_k^e with 1<=e<=log_p_k p_n#, and their multiples with primes p_T < q < p_T#. All numbers in these columns are negative. n r(p_n#) 2 3 5 7 11 13 17 19 23 29 -------------------------------------------------------------------------------- 1| 2 1 2| 5 3 2 3| 18 11 9 6 4| 68 47 37 27 22 5| 283 206 170 130 109 84 6| 1161 871 734 583 500 405 373 7| 4843 3732 3190 2601 2268 1877 1746 1556 8| 19985 15680 13554 11239 9906 8337 7813 7031 6728 9| 83074 66141 57713 48461 43107 36738 34601 31405 30155 28111 10| 349670 281949 248018 210504 188649 162498 153691 140428 135239 126722 117035 11| 12| This table shows the "damage" incurred if we were able to "leave out" prime p_k and then assess r(p_n#). The bound on the infinite regular tensor I is held constant at p_n# and one distinct prime divisor p_k and any regular product with p_k as a factor are simply ignored. For example, if we were to ignore the prime 2, its powers, and the multiples of p with all totatives t (t such that GCD(t,p_n#)=1) for p_3# = 30, r'(30) would be 11 less than r(30), i.e., 7. We can see this by looking at the regulars of 30 and eliminating those involving 2: x x x x x x x x x x x = 11 x's {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30} = 7 regulars left over. If we were to ignore the prime 3 we would see the following: x x x x x x x x x = 9 x's {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30} = 9 regulars left over. Finally ignoring the prime 5: x x x x x x 6 x's {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30} = 12 regulars left over. It stands to reason that ignoring smaller prime divisors p generates a larger number in Table 6 and a smaller partial count r'(n) in Table 7, since log_p n is larger for smaller primes p. This is a fuller indication that implies small prime totatives exert the heaviest downward force against a higher regular counting function. Table 7. Partial count of the regulars of p_n#, r'_k (p_n#), leaving out those that have p_k as a factor. This table was produced by subtracting the corresponding values in Table 6 from r(p_n#). n r(p_n#) 2 3 5 7 11 13 17 19 23 29 31 --------------------------------------------------------------------------------------- 1| 2 1 2| 5 2 3 3| 18 7 9 12 4| 68 21 31 41 46 5| 283 77 113 153 174 199 6| 1161 290 427 578 661 756 788 7| 4843 1111 1653 2242 2575 2966 3097 3287 8| 19985 4305 6431 8746 10079 11648 12172 12954 13257 9| 83074 16933 25361 34613 39967 46336 48473 51669 52919 54963 10| 349670 67721 101652 139166 161021 187172 195979 209242 214431 222948 232635 11|1456458 270340 406560 557949 646576 753176 789249 843679 865040 900169 940261 951295 The final values in the above table, i.e., the diagonal 1-3-12-46-..., comprises values that r(p_T#) would have if we ignored p_T and is the basis of comparison for the "Phantom Jump Ratio" (PJR). It's important to keep in mind that the adjusted figures in Table 7 pertain to no real number, or rather are only a partial count of the regulars of p_n# leaving out those that have p_k as a factor. Table 8. Ratios of Table 7 figures r'(p_n#, k)/r(p_n#). n r(p_n#) 2 3 5 7 11 13 17 19 23 29 31 --------------------------------------------------------------------------------------- 1| 2 2 2| 5 2.5 1.6667 3| 18 2.5714 2 1.5 4| 68 3.2381 2.1936 1.6585 1.4783 5| 283 3.6753 2.5044 1.8497 1.6264 1.4221 6| 1161 4.0035 2.7190 2.0087 1.7564 1.5357 1.4734 7| 4843 4.3591 2.9298 2.1601 1.8808 1.6328 1.5638 1.4734 8| 19985 4.6423 3.1076 2.2850 1.9828 1.7158 1.6419 1.5428 1.5075 9| 83074 4.9060 3.2757 2.4001 2.0786 1.7929 1.7138 1.6078 1.5698 1.5115 10| 349670 5.1634 3.4399 2.5126 2.1716 1.8682 1.7842 1.6711 1.6307 1.5684 1.5031 11|1456458 5.3875 3.5824 2.6104 2.2526 1.9338 1.8454 1.7263 1.6837 1.6180 1.5490 1.5310 Table 9. Let r'(n,k) be the number of regulars of p_n# when k=0, the number of regulars ignoring prime p_(n+k+1) when k is negative, and ignoring p_n but considering p_(n+k) when k is positive. Note that r'(n,0) = r(p_n#). The cases of negative k illustrate that r'(n) is greatest as we ignore larger prime factors p_(n+k+1) | p_n#, but r(p_n#) > r'(n,-k). The cases of positive k illustrate that r'(n) is greatest for p_(n-1)# p(n+k) with k=1, but smaller than r(p_n#). <------ number of primes = n-1 | number of primes = n ------> n -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 ---------------------------------------------------------------------------------------------- 1| 1 2 1 1 1 1 2| 2 3 5 4 3 3 3 3| 7 9 12 18 16 14 14 13 4| 21 31 41 46 68 60 59 56 55 5| 77 113 153 174 199 283 273 260 255 249 6| 290 427 578 661 756 788 1161 1100 1078 1045 1011 7| 1111 1653 2242 2575 2966 3097 3287 4843 4739 4584 4424 4382 8| 4305 6431 8746 10079 11648 12172 12954 13257 19985 19300 18583 18394 17937 9| 25361 34613 39967 46336 48473 51669 52919 54963 83074 79854 79017 76965 75869 10| 139166 161021 187172 195979 209242 214431 222948 232635 349670 345867 336500 331523 329308 +-------+ | Code. | +-------+ This space reserved to share Wolfram code to generate the sequence once the paper is out, and the goal of research is reached. All code written by Michael Thomas De Vlieger between February 2015 and May 2017, see notes where indicated. Some code requires at least Mathematica 10.2 because of MissingQ, associations. r(n) = A010846(n): regular counting functions: r[n_] := Function[d, Length@ Prepend[ Select[Range[2, n], SubsetQ[d, FactorInteger[#][[All, 1]]] &], 1]][FactorInteger[n][[All, 1]]] (* prime divisor subset PDS approach *) r[n_] := Select[Range@ n, First@ NestWhile[Function[s, {#1/s, s}]@ GCD[#1, #2] & @@ # &, {#, n}, And[First@ # != 1, ! CoprimeQ @@ #] &] == 1 &] (* Recursive GCD approach *) r[n_] := Count[Range@n, k_ /; First@ NestWhile[Function[s, {#1/s, s}]@ GCD[#1, #2] & @@ # &, {k, n}, And[First@ # != 1, ! CoprimeQ @@ #] &] == 1] (* Recursive GCD approach *) r[n_] := Count[Range@ n, k_ /; PowerMod[n, k, k] == 0] (* Congruency test: k | n^k *) r[n_] := With[{m = n^Floor@ Log2@ n}, Count[Range@ n, k_ /; Divisible[m, k]]](* Congruency test: k | n^floor(log_2 n) *) r[n_] := Total@ Map[MoebiusMu[#] Floor[n/#] &, Select[Range[n - 1], CoprimeQ[#, n] &]] (* Mertens Function across totatives of n *) Most efficient r(n): O(log n) time: r[n_] := Length@ Function[w, ToExpression@ StringJoin["Module[{n = ", ToString@ n, ", k = 0}, Flatten@ Table[k++, ", Most@ Flatten@ Map[{#, ", "} &, #], "]]"] &@ MapIndexed[ Function[p, StringJoin["{", ToString@ Last@ p, ", 0, Log[", ToString@ First@ p, ", n/(", ToString@ InputForm[ Times @@ Map[Power @@ # &, Take[w, First@ #2 - 1]]], ")]}"]]@ w[[First@ #2]] &, w] ]@ Map[{#, ToExpression["p" <> ToString@ PrimePi@ #]} &, FactorInteger[n][[All, 1]]] (* Regular Count | Logarithmic Constructor Approach | 201701101605 (p_8# in 1s, p_9# in 4.625s, p_10# in 20s, p_11# in 83.5s, p_12# in 355s, p_13# in 1530s (25.5m), p_14# in (110m), p_15# in (7:52:30) h, p_16# in (33:51) h) *) r[n_] := Function[w, ToExpression@ StringJoin["Module[{k = 0, n = ", ToString@ n, "}, ", StringDrop[#, -1], "; k]"] &@ StringJoin@ Fold[{StringJoin["Do[", First@ #1, ToString@ #2, "],"], StringJoin[Last@ #1]} &, {"k++, ", ""}, #] &@ Reverse@ MapIndexed[ Function[p, StringJoin["{", ToString@ Last@ p, ", 0, Log[", ToString@ First@ p, ", n/(", ToString@ InputForm[ Times @@ Map[Power @@ # &, Take[w, First@ #2 - 1]]], ")]}"]]@ w[[First@ #2]] &, w] ]@ Map[{#, ToExpression["p" <> ToString@PrimePi@#]} &, FactorInteger[n][[All, 1]]] (* Regular Count | Logarithm Approach | Do Count | 201701131240 (p_8# in 1.33 s, p_9# in 5.55 s, p_10# in 23.5s *) Compute r(n) for all n with the same squarefree root - very efficient: levelwise[T_, k1_: 1, k2_: 1, m_: 1] := Module[{P = Times @@ Prime@ Range@ T, n = If[k2 == 1, NextPrime@ Prime@ T - 1, Min[k2, NextPrime@ Prime@ T - 1]], x, r}, x = If[m == 1, P, m]; r[x_] := Function[w, ToExpression@ StringJoin["With[{n=", ToString@ x, "}, Flatten@ Table[", ToString@ InputForm[Times @@ Map[Power @@ # &, w]], ", ", Most@ Flatten@ Map[{#, ", "} &, #], "]]"] &@ MapIndexed[ Function[p, StringJoin["{", ToString@ Last@ p, ", 0, Log[", ToString@ First@ p, ", n/(", ToString@ InputForm[ Times @@ Map[Power @@ # &, Take[w, First@ #2 - 1]]], ")]}"]]@ w[[First@ #2]] &, w] ]@ Map[{#, ToExpression["p" <> ToString@ PrimePi@ #]} &, FactorInteger[x][[All, 1]]]; Function[w, Map[Function[k, {k x, Length@ TakeWhile[w, # <= k x &]}], Range[k1, n]]]@ Sort@ r[x n]] (* Segmented Levelwise Regular Count kx | 201701151118 (k p_8# in 3.63s, k p_9# in 16.5s, k p_10# in 73s, k p_11# in 325.5s (5:25.5m), k p_12# in 1486s (25m), k p_13# in (6220s 1:53h), k p_14# in 8:44h) *) primorialP[n_] := Times @@ Prime@ Range@ n; (* A002110(n) *) encode[n_] := If[n == 1, {0}, Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ n] (* "Multiplicity notation": Reverse@ IntegerDigits@ A054841(n) *); decode[n_] := Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, n]; (* Reverse@ IntegerDigits@ A054841(n) -> decimal. *) picode[n_Integer] := Prepend[Differences@#, First@#] &@ Flatten[FactorInteger[n] /. {p_, e_} /; p > 0 :> ConstantArray[PrimePi@p, e]]; (* "pi-difference notation": see OEIS A287352. *) pidecode[w_List] := Times @@ Map[If[# == 0, 1, Prime@#] &, Accumulate@w]; (* OEIS A287352 -> decimal. *) zerocode[n_] := Prepend[Differences@#, First@#] - 1 &@ PrimePi@FactorInteger[n][[All, 1]]; (* "zerocode": run lengths of indices of prime totatives, essentially OEIS A287352(n) - 1. *) fromZerocode[w_, n_: 0] := Block[{nu = If[n == 0, Length@w, n - 1]}, If[w == {}, 2, Times @@ Flatten@MapIndexed[Prime[#2]^#1 &, #] &@ If[Last@# == 0, Join[#, {1}], #] &@ Flatten[# /. a_ /; Depth@a == 2 :> If[First@a == 0, ConstantArray[1, Length@a + 1], Most@Riffle[#, ConstantArray[1, Length@#]] &@ Map[ConstantArray[0, #] &, a]]] &@ SplitBy[PadLeft[w, nu], # == 0 &]]]; (* zerocode -> decimal. *) sweep[n_] := Block[{primorialP, jDepth, jFrustum, encode, decode}, primorialP[x_] := Times @@ Prime@ Range@ x; jDepth[ x_] := -1 + SelectFirst[Range@ x, Function[j, # Prime[x + 1]/Prime[x - j + 1] > 2 #] &[ Times @@ Prime@ Range@ x]] /. j_ /; MissingQ@ j -> 1; tierGCD[x_] := primorialP[ x - # + 1] &@(SelectFirst[Range@ x, Function[k, # Prime[x + 1]/Prime[x - k + 1] > 2 #] &@ primorialP@ x] /. k_ /; MissingQ@ k -> 1); encode[x_] := If[x == 1, {0}, Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ x]; decode[x_] := Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, x]; Map[Select[#, Function[k, # <= k <= 2 # &@ primorialP@ n]] &, {{primorialP[n]}}~Join~ Map[Function[j, Function[i, Map[decode@ Join[encode@ tierGCD@ n, ConstantArray[1, jDepth@ n - j], {0}, Reverse@ IntegerDigits@ FromDigits@ #] &, Reverse@ Rest@ Permutations[ ConstantArray[1, j]~Join~ ConstantArray[0, i], {i + j}]]][ If[# == 1, 1, PrimePi[2 Prime@ #]] - n] &[n - j + 1]], Range@ jDepth@ n]]] (* sweep | 201609142100 | output all level-1 candidates for A244052 including A060735 terms. Permutations based. *); sweep[n_, l_: 0, o_: 0] := If[n == 0, {{1}}, Block[{P = Product[Prime@ i, {i, n}], lim, k = 1, m = 0, c, w = ConstantArray[1, n]}, lim = If[l <= 1, Prime[n + 1] P, l P]; If[o <= 0, {P}, {w}]~Join~ Reap[If[o <= 0, Do[w = If[k == 1, MapAt[# + 1 &, w, -k], Join[Drop[MapAt[# + 1 &, w, -k], -k + 1], ConstantArray[1, k - 1]]]; c = Times @@ Map[If[# == 0, 1, Prime@ #] &, Accumulate@ w]; If[c < lim, Sow[c]; k = 1, If[k == n, Break[], k++]], {i, Infinity}], Do[ w = If[k == 1, MapAt[# + 1 &, w, -k], Join[Drop[MapAt[# + 1 &, w, -k], -k + 1], ConstantArray[1, k - 1]]]; c = Times @@ Map[If[# == 0, 1, Prime@ #] &, Accumulate@ w]; If[c < lim, Sow[w]; k = 1, If[k == n, Break[], k++]], {i, Infinity}] ] ][[-1, 1]] ]] (* loopsweep | 201705251335 | output all squarefree numbers p_n# <= m < lim, with omega(m) = n, and adjustable lim = k*p_n# or p_(n+1). Most efficient; loop based. Can output n = 100 for lim = 2 p_n#. Can output decimal m or OEIS A287352(m); uses A287352 to generate m. *); countsweep[n_, l_: 0] := If[n == 0, 1, Block[{P = Product[Prime@i, {i, n}], lim, k = 1, m = 0, a, c = 1, w = ConstantArray[1, n]}, lim = If[l <= 1, Prime[n + 1] P, l P]; Do[w = If[k == 1, MapAt[# + 1 &, w, -k], Join[Drop[MapAt[# + 1 &, w, -k], -k + 1], ConstantArray[1, k - 1]]]; a = Times @@ Map[If[# == 0, 1, Prime@#] &, Accumulate@w]; If[a < lim, c++; k = 1, If[k == n, Break[], k++]], {i, Infinity}]; c ] ] (* coutsweep | 201705251650 | count all squarefree numbers p_n# <= m < lim, with omega(m) = n, and adjustable lim = k*p_n# or p_(n+1). Most efficient; loop based on A287352. *); tier[n_] := Sort@ Flatten@ Map[Function[k, DeleteCases[ Map[Select[#, # < (k + 1) primorialP@ n &] &, (k sweep[n])], {}]], Range[Prime[n + 1] - 1]] (* Permutations-based sweep: generate all candidates in tier n *); tier[n_] := Sort@ Flatten@ Map[Function[k, DeleteCases[ Select[k sweep[n, 2], # < (k + 1) primorialP@ n &], {}]], Range[Prime[n + 1] - 1]]; (* loop-sweep: generate all candidates in tier n: faster *); jDepth[n_] := -1 + SelectFirst[Range@ n, Function[k, # Prime[n + 1]/Prime[n - k + 1] > 2 #] &[ Times @@ Prime@ Range@ n]] /. k_ /; MissingQ@ k -> 1 (* algebraic j-depth of primorial(omega) *); jNumber[n_] := Block[{primorialP, j}, primorialP[x_] := Times @@ Prime@ Range@ x; j = (SelectFirst[Range@ n, Function[k, # Prime[n + 1]/Prime[n - k + 1] > 2 #] &@ primorialP@ n] /. k_ /; MissingQ@ k -> 1) - 1; Floor[primorialP[n + 1]/Prime[n - j + 1]]] (* algebraic deepest tier-T turbulent number m = number with least prime totative lpt of tier *); tierGCD[n_] := Block[{primorialP, j}, primorialP[x_] := Times @@ Prime@ Range@ x; j = (SelectFirst[Range@ n, Function[k, # Prime[n + 1]/Prime[n - k + 1] > 2 #] &@ primorialP@ n] /. k_ /; MissingQ@ k -> 1) - 1; primorialP[n - j]] (* algebraic j-depth of primorial(T) = GCD of tier, formerly "jFrustum" *); primeFrustum[n_] := n + 1 - SelectFirst[Range@ n, Function[k, # Prime[n + 1]/Prime[n - k + 1] > 2 #] &[ Times @@ Prime@ Range@ n]] /. k_ /; MissingQ@ k -> 1 (* primorial frustum at depth of primorial(omega) = pi(gpf(GCD of tier T))*) iDistension[n_, j_] := -1 + SelectFirst[Range@ n, Function[i, # Prime[n + i]/Prime[n - j + 1] > 2 #] &[ Times @@ Prime@ Range@ n]] /. k_ /; MissingQ@ k -> 1 (* Distension of primorial(n) at depth j, i.e., greatest pi(gpf) for numbers m with lpt=prime(T-j+1) *); iNumber[n_, j_] := Block[{primorialP, i}, primorialP[x_] := Times @@ Prime@ Range@ x; i = (SelectFirst[Range@ n, Function[k, # Prime[n + k]/Prime[n - j + 1] > 2 #] &@ primorialP@ n] /. k_ /; MissingQ@ k -> 1) - 1; If[i == 0, 1, primorialP[n - j] (Times @@ Prime@ Range[n - j + 2, n]) Prime[n + i]]] (* algebraic most distended omega-turulent candidate at depth j, i.e., m with omega(m)=T that has greatest gpf with lpt at prime(T-j+1) *) f[T_] := {{T, primorialP@T}, Map[Function[ w, {StringReverse@ ToString@ FromDigits@ Reverse@ Take[#, -(Length@ # - Length@ TakeWhile[#, # == 1 &])] &@ encode@ w (* abbreviate notation *), #, Lookup[t, #]} &@ w] /@ Select[#, Function[k, # <= k <= 2 # &@ primorialP[T]]] &, Map[Function[j, Function[i, Map[Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, #](* decode *)&@ Join[encode@ tierGCD@ T, ConstantArray[1, jDepth@ T - j], {0}, Reverse@ IntegerDigits@ FromDigits@ #] &, Reverse@ Rest@ Permutations[ ConstantArray[1, j]~Join~ConstantArray[0, i], {i + j}]]][ If[# == 1, 1, PrimePi[2 Prime@ #]] - T] &[T - j + 1]], Range@ jDepth@ T]]} (* Analyze Tier T turbulent terms by depth class: Format: abbreviated notation - n - r(n) - Presence in A244052/3 *) attributableToGPF[n_] := Function[P, Total@ Map[MoebiusMu[#] Floor[P/#] &, Select[Range@ P, And[CoprimeQ[#, Times @@ Prime@ Range[n - 1]], Divisible[#, Prime@ n]] &]]][Times @@ Prime@ Range@ n] (* portion of r(p_n#) attributable to p_n 201702071500, based on Mertens function *) attributableToPrimek[n_, k_] := Function[P, Total@ Map[MoebiusMu[#] Floor[P/#] &, Select[Range@ P, And[CoprimeQ[#, Times @@ Prime@ Range@ n/Prime@ k], Divisible[#, Prime@ k]] &]]][Times @@ Prime@ Range@ n] (* portion of r(p_n#) attributable to p_k with 1<=k<=n 201702072030 *) attributableToPrimek[n_, k_] := Block[{P = Times @@ Prime@ Range@ n}, Length@ Function[w, ToExpression@ StringJoin["Module[{n = ", ToString@P, ", k = 0}, Flatten@ Table[k++, ", Most@ Flatten@ Map[{#, ", "} &, #], "]]"] &@ MapIndexed[ Function[p, StringJoin["{", ToString@Last@p, ", 0, Log[", ToString@ First@ p, ", n/(", ToString@ InputForm[ Times @@ Map[Power @@ # &, Take[w, First@ #2 - 1]]], ")]}"]]@ w[[First@ #2]] &, w] ]@ Map[{#, ToExpression["p" <> ToString@ PrimePi@ #]} &, FactorInteger[P/Prime@ k][[All, 1]]] ] rPrime[n_, k_] := Block[{P = Times @@ Prime@ Range@ n, m = Which[n + k < 0, Times @@ Prime@ Range@ n, k > 0, Prime[n]/(Prime[n + k]), k < 0, Prime[n + k + 1], True, 1]}, If[n + k < 0, 0, Length@ Function[w, ToExpression@ StringJoin["Module[{n = ", ToString@ P, ", k = 0}, Flatten@ Table[k++, ", Most@ Flatten@ Map[{#, ", "} &, #], "]]"] &@ MapIndexed[ Function[p, StringJoin["{", ToString@ Last@ p, ", 0, Log[", ToString@ First@ p, ", n/(", ToString@ InputForm[ Times @@ Map[Power @@ # &, Take[w, First@ #2 - 1]]], ")]}"]]@ w[[First@ #2]] &, w] ]@ Map[{#, ToExpression["p" <> ToString@ PrimePi@ #]} &, FactorInteger[P/m][[All, 1]]] ]] (* r'(n) | Logarithmic Constructor Approach | 201703061030 *) Some interesting runs: 1. Generate candidates and multiplicity notations for the candidates in tier T: {#, FromDigits@ encode@ #} & /@ tier@ 5 // TableForm 2. Generate deepest candidate m (with maximum j) in tier T: {primorialP@ #, jDepth@ #, FromDigits@ encode@ jNumber@ #, 2 primorialP@ #} & /@ Range@ 8 // TableForm {primorialP@ #, jNumber@ #, FromDigits@ encode@ jNumber@ #, tierGCD@ #, primeFrustum@ #, FromDigits@ encode[Times @@ Prime@ Range@ primeFrustum@ #], 2 primorialP@ #} & /@ Range@ 12 // TableForm 3. Compare cardinal distension to tier GCD (cardinal depth) for tier T: {#, # + iDistension[#, 1], primeFrustum@ #} & /@ Range@ 36 // TableForm 4. Generate Table 1: Table[Map[iDistension[n, #] &, Range@ jDepth@ n] /. {} -> {0}, {n, 40}] // TableForm (* T(n,k) with n = omega, k = distension at depth = position(k) *) With[{n = 12}, {#, FromDigits@ encode@ # & /@ Map[iNumber[n, #] &, Range@ jDepth @n] /. {} -> {0}, 2 #} &@ primorialP @ n] (* deepest j, most distended i for j *) 5. Generate multiplicity notations present in tier T sorted by depth: Table[{primorialP@ n, FromDigits[encode@#] & /@ Select[#, Function[k, # <= k <= 2 # &@ primorialP[n]]] & /@ Map[Function[j, Function[i, Map[Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, #](* decode *)&@ Join[encode@ tierGCD@ n, ConstantArray[1, jDepth@ n - j], {0}, Reverse@ IntegerDigits@ FromDigits@ #] &, Reverse@ Rest@ Permutations[ ConstantArray[1, j]~Join~ ConstantArray[0, i], {i + j}]]][ If[# == 1, 1, PrimePi[2 Prime@ #]] - n] &[n - j + 1]], Range@ jDepth@ n]}, {n, 1, 15}] // TableForm (* signatures per class 20161023 *) 6. Generate Table 3: Table[Length@ Select[#, Function[k, # <= k <= 2 # &@ primorialP[n]]] & /@ Map[Function[j, Function[i, Map[Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, #](* decode *)&@ Join[encode@ tierGCD@ n, ConstantArray[1, jDepth@ n - j], {0}, Reverse@ IntegerDigits@ FromDigits@ #] &, Reverse@ Rest@ Permutations[ ConstantArray[1, j]~Join~ConstantArray[0, i], {i + j}]]][ If[# == 1, 1, PrimePi[2 Prime@ #]] - n] &[n - j + 1]], Range@ jDepth@ n], {n, 2, 32}] // TableForm (* Sweep Program | Enumerator 20161023 *) Table[Length /@ SplitBy[#, Length] &@ Map[Drop[#, Length@ TakeWhile[#, # <= 1 &]] &, Rest@ sweep[n, 2, 1]], {n, 1, 60}] // TableForm (* Loopsweep Program | Enumerator 20170529 | 1 <= T <= 60 in about 1 minute *) 7. Generate Table 6: Table[(-1) attributableToPrimek[n, k], {n, 8}, {k, n}] // TableForm 8. Generate Table 7 (This may be more efficient via memoization in a Do loop): Table[r[Times @@ Prime@ Range@ n] + attributableToPrimek[n, k], {n, 8}, {k, n}] // TableForm 9. Generate Table 9: Table[rPrime[n, k], {n, 2, 10}, {k, -n, 14 - n}] // TableForm End of code section. +-----------------+ | Update History. | +-----------------+ Most recent updates: Updated 201702041200 to include all original code written toward the problem. Updated 201705300800: Added loopsweep-related content. Tables 1, 3, and 5 extended. Last updated 201706191745: Added zerocode abbreviations to turbulent terms in Data, extended table 5, explained zerocode and pi-code. Added term 3 to the data and re-indexed a(n). +-------------------+ | Acknowledgements. | +-------------------+ Thanks to David Corneth for computation of terms 55-149 and primorials. Love and thanks to my wife Laura for supporting my mathematics endeavors, even though I am by training and profession an architect. This work is dedicated to the memory of Ms. Mary Svitek, my middle school math teacher, and to my aunt Mary Katherine (De Vlieger) Nuzzo, who encouraged me in math throughout childhood. Aunt Kath was a "woman in STEM" (computer science) in the 1960s, long before it was "a thing". Without the encouragement of these women, perhaps I would not have a love for mathematics. Written and researched in Southampton Neighborhood in the city of St. Louis, Missouri, USA. (EOF End of file.)