# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a306737 Showing 1-1 of 1 %I A306737 #17 Mar 08 2019 20:23:42 %S A306737 0,1,1,1,2,1,2,1,1,2,2,2,1,1,1,2,1,3,1,1,3,2,3,1,1,1,3,1,2,3,1,1,2,3, %T A306737 1,1,4,2,4,1,1,1,4,1,2,4,1,1,2,4,2,2,4,1,1,1,2,4,1,2,2,4,1,1,1,1,2,4, %U A306737 1,1,3,4,1,2,5,2,2,2,4,1,1,1,3,4,1,1,2,5,2,2,5,1,1,1,2,5,1,2,2,5,1,1,1,1,2,5 %N A306737 Irregular triangle where row n is a list of indices in A002110 with multiplicity whose product is A002182(n). %C A306737 Each highly composite number A002182(n) can be expressed as a product of primorials in A002110. %C A306737 Row 1 = {0} by convention. %C A306737 Maximum value in row n is given by A001221(A002182(n)). %C A306737 Row n in reverse order is the conjugate of A067255(A002182(n)), a list of the multiplicities of the prime divisors of A002182(n). %H A306737 Michael De Vlieger, Table of n, a(n) for n = 1..10198, rows 1 <= n <= 1200, flattened. %H A306737 Michael De Vlieger, Charts showing terms in A002182 as a product of terms in A002110. %H A306737 Michael De Vlieger, Condensed text table showing terms in rows 1 <= n <= 10000. %H A306737 Benny Lim, Prime Numbers Generated From Highly Composite Numbers, Parabola (2018) Vol. 54, Issue 3. %e A306737 Terms in the first rows n of this sequence, followed by the corresponding primorials whose product = A002182(n): %e A306737 n T(n,k) A002110(T(n,k)) A002182(n) %e A306737 ----------------------------------------------- %e A306737 1: 0; 1 = 1 %e A306737 2: 1; 2 = 2 %e A306737 3: 1, 1; 2 * 2 = 4 %e A306737 4: 2; 6 = 6 %e A306737 5: 1, 2; 2 * 6 = 12 %e A306737 6: 1, 1, 2; 2 * 2 * 6 = 24 %e A306737 7: 2, 2; 6 * 6 = 36 %e A306737 8: 1, 1, 1, 2; 2 * 2 * 2 * 6 = 48 %e A306737 9: 1, 3; 2 * 30 = 60 %e A306737 10: 1, 1, 3; 2 * 2 * 30 = 120 %e A306737 11: 2, 3; 6 * 30 = 180 %e A306737 12: 1, 1, 1, 3; 2 * 2 * 2 * 30 = 240 %e A306737 13: 1, 2, 3; 2 * 6 * 30 = 360 %e A306737 14: 1, 1, 2, 3; 2 * 2 * 6 * 30 = 720 %e A306737 15: 1, 1, 4; 2 * 2 * 210 = 840 %e A306737 ... %e A306737 Row 6 = {1,1,2} since A002110(1)*A002110(1)*A002110(2) = 2*2*6 = 24 and A002182(6) = 24. The conjugate of {2,1,1} = {3,1} and 24 = 2^3 * 3^1. %e A306737 Row 10 = {1,1,3} since A002110(1)*A002110(1)*A002110(3) = 2*2*30 = 120 and A002182(10) = 120. The conjugate of {3,1,1} = {3,1,1} and 120 = 2^3 * 3^1 * 5^1. %t A306737 With[{s = DivisorSigma[0, Range[250000]]}, Map[Reverse@ Table[LengthWhile[#, # >= i &], {i, Max@ #}] &@ If[# == 1, {0}, Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ #] &@ FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]] /. {} -> {0}] // Flatten %Y A306737 Cf. A001221, A002110, A002182, A067255, A304886. %K A306737 nonn,tabf %O A306737 1,5 %A A306737 _Michael De Vlieger_, Mar 06 2019 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE