+-----------------------------------------------------+ | Maximum "Distension" i Given "Tier" m and "Depth" j | +-----------------------------------------------------+ Written by Michael Thomas De Vlieger, St. Louis, Missouri, USA, 20170626. a(n) = A288813(n) = Irregular triangle read by rows: T(m,k) is the list of numbers A002110(m) <= t < 2*A002110(m) such that A001221(t) = m. a(n) produces the terms k*t with k = 1 in A288784. In this work, m is called a "tier" or row of irregular triangle T(m,k). All terms of a(n) are deemed "level 1" and "primary turbulent" according to the document https://oeis.org/A288784/a288784.txt. Let the numbers a(n) = t = {3, 10, 42, 330, 390, 2730, 3570, 3990, 4290, ...} be "turbulent" numbers on account of the aggregate appearance of A054841(a(n)) in A288784. These numbers are "turbulent" because pi(A006530(a(n))) > m. +-------+ | DATA. | +-------+ "n" is the position in a(n) = n, (position of A288813(n) in A288813). "m" is the index of the largest primorial p_m# = A002110(m) < a(n). All terms of p_m# <= a(n) < p_(m+1)# must have A001221(a(n)) = m. "j": "Depth" j = n - pi(A053669(A288784(x))) + 1. This is the position of first "0" in A054841(a(n)) minus m plus 1. The variable j is descriptive of the greatest common divisor A002110(j - 1) of all terms of T(m,k) for a given tier m. "i": "Distension" i = pi(A006530(A288784(x))) - n. This is the length of A054841(a(n)) minus m. The variable i is descriptive of the greatest prime factor of a term t with j set to a particular value. Turbulent terms t (i.e., all terms of a(n)) have distension i > 0 and depth j > 0. Reading the table diagonally downward to the right, one observes that the vaule decrements by 1 from the first column's figure. Table 1. Distension i at depth j (horizontal axis) in tier m (vertical axis): 1| 0 2| 1 3| 1 4| 2 5| 3 1 6| 3 2 7| 4 2 1 8| 4 3 1 9| 5 3 2 10| 6 4 2 1 11| 7 5 3 1 12| 9 6 4 2 13| 9 8 5 3 1 14| 9 8 7 4 2 15| 9 8 7 6 3 1 16| 11 8 7 6 5 2 17| 13 10 7 6 5 4 1 18| 12 12 9 6 5 4 3 19| 13 11 11 8 5 4 3 2 20| 14 12 10 10 7 4 3 2 1 21| 13 13 11 9 9 6 3 2 1 22| 15 12 12 10 8 8 5 2 1 23| 15 14 11 11 9 7 7 4 1 24| 16 14 13 10 10 8 6 6 3 25| 19 15 13 12 9 9 7 5 5 2 26| 20 18 14 12 11 8 8 6 4 4 1 27| 19 19 17 13 11 10 7 7 5 3 3 28| 19 18 18 16 12 10 9 6 6 4 2 2 29| 18 18 17 17 15 11 9 8 5 5 3 1 1 m 30| 18 17 17 16 16 14 10 8 7 4 4 2 31| 23 17 16 16 15 15 13 9 7 6 3 3 1 32| 23 22 16 15 15 14 14 12 8 6 5 2 2 33| 25 22 21 15 14 14 13 13 11 7 5 4 1 1 34| 25 24 21 20 14 13 13 12 12 10 6 4 3 35| 27 24 23 20 19 13 12 12 11 11 9 5 3 2 36| 26 26 23 22 19 18 12 11 11 10 10 8 4 2 1 37| 28 25 25 22 21 18 17 11 10 10 9 9 7 3 1 38| 28 27 24 24 21 20 17 16 10 9 9 8 8 6 2 39| 28 27 26 23 23 20 19 16 15 9 8 8 7 7 5 1 40| 28 27 26 25 22 22 19 18 15 14 8 7 7 6 6 4 41| 30 27 26 25 24 21 21 18 17 14 13 7 6 6 5 5 3 42| 30 29 26 25 24 23 20 20 17 16 13 12 6 5 5 4 4 2 43| 32 29 28 25 24 23 22 19 19 16 15 12 11 5 4 4 3 3 1 44| 32 31 28 27 24 23 22 21 18 18 15 14 11 10 4 3 3 2 2 45| 32 31 30 27 26 23 22 21 20 17 17 14 13 10 9 3 2 2 1 1 46| 32 31 30 29 26 25 22 21 20 19 16 16 13 12 9 8 2 1 1 47| 35 31 30 29 28 25 24 21 20 19 18 15 15 12 11 8 7 1 48| 38 34 30 29 28 27 24 23 20 19 18 17 14 14 11 10 7 6 49| 38 37 33 29 28 27 26 23 22 19 18 17 16 13 13 10 9 6 5 50| 38 37 36 32 28 27 26 25 22 21 18 17 16 15 12 12 9 8 5 4 51| 39 37 36 35 31 27 26 25 24 21 20 17 16 15 14 11 11 8 7 4 3 52| 39 38 36 35 34 30 26 25 24 23 20 19 16 15 14 13 10 10 7 6 3 2 53| 39 38 37 35 34 33 29 25 24 23 22 19 18 15 14 13 12 9 9 6 5 2 1 54| 41 38 37 36 34 33 32 28 24 23 22 21 18 17 14 13 12 11 8 8 5 4 1 55| 42 40 37 36 35 33 32 31 27 23 22 21 20 17 16 13 12 11 10 7 7 4 3 56| 43 41 39 36 35 34 32 31 30 26 22 21 20 19 16 15 12 11 10 9 6 6 3 2 57| 42 42 40 38 35 34 33 31 30 29 25 21 20 19 18 15 14 11 10 9 8 5 5 2 1 58| 42 41 41 39 37 34 33 32 30 29 28 24 20 19 18 17 14 13 10 9 8 7 4 4 1 59| 42 41 40 40 38 36 33 32 31 29 28 27 23 19 18 17 16 13 12 9 8 7 6 3 3 60| 42 41 40 39 39 37 35 32 31 30 28 27 26 22 18 17 16 15 12 11 8 7 6 5 2 2 ----------------------------------------------------------------------------------------------------------- 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 "depth" j Table 2. Distension i for prime q (horizontal axis) in tier m (vertical axis), a transform of the above table. This transform makes it clear that the values derive from a simple formula related to A020900: m p_m# 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 <- pi(q) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 <- prime q -- 末 末 末 末 末 末 末 末 末 末 末 末 末 -- -- -- 1 2# 0 2 3# 1 3 5# 1 4 7# 2 5 11# 1 3 6 13# 2 3 7 17# 1 2 4 8 19# 1 3 4 9 23# 2 3 5 10 29# 1 2 4 6 11 31# 1 3 5 7 12 37# 2 4 6 9 13 41# 1 3 5 8 9 14 43# 2 4 7 8 9 15 47# 1 3 6 7 8 9 16 53# 2 5 6 7 8 11 17 59# 1 4 5 6 7 10 18 61# 3 4 5 6 9 19 67# 2 3 4 5 8 20 71# 1 2 3 4 7 21 73# 1 2 3 6 22 79# 1 2 5 23 83# 1 4 24 89# 3 25 97# 2 26 101# 1