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A356234 Irregular triangle read by rows where row n is the ordered factorization of n into maximal gapless divisors.

%I A356234 #5 Aug 30 2022 09:41:31
%S A356234 2,3,4,5,6,7,8,9,2,5,11,12,13,2,7,15,16,17,18,19,4,5,3,7,2,11,23,24,
%T A356234 25,2,13,27,4,7,29,30,31,32,3,11,2,17,35,36,37,2,19,3,13,8,5,41,6,7,
%U A356234 43,4,11,45,2,23,47,48,49,2,25,3,17,4,13,53,54,5,11,8
%N A356234 Irregular triangle read by rows where row n is the ordered factorization of n into maximal gapless divisors.
%C A356234 Row-products are the positive integers 1, 2, 3, ...
%e A356234 The first 16 rows:
%e A356234    1 =
%e A356234    2 = 2
%e A356234    3 = 3
%e A356234    4 = 4
%e A356234    5 = 5
%e A356234    6 = 6
%e A356234    7 = 7
%e A356234    8 = 8
%e A356234    9 = 9
%e A356234   10 = 2 * 5
%e A356234   11 = 11
%e A356234   12 = 12
%e A356234   13 = 13
%e A356234   14 = 2 * 7
%e A356234   15 = 15
%e A356234   16 = 16
%e A356234 The factorization of 18564 is 18564 = 12*7*221, so row 18564 is {12,7,221}.
%t A356234 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A356234 Table[Times@@Prime/@#&/@Split[primeMS[n],#1>=#2-1&],{n,100}]
%Y A356234 Row-lengths are A287170, firsts A066205, even bisection A356229.
%Y A356234 Applying bigomega to all parts gives A356226, statistics A356227-A356232.
%Y A356234 A001055 counts factorizations.
%Y A356234 A001221 counts distinct prime factors, sum A001414.
%Y A356234 A003963 multiplies together the prime indices.
%Y A356234 A056239 adds up the prime indices, row sums of A112798.
%Y A356234 A132747 counts non-isolated divisors, complement A132881.
%Y A356234 A356069 counts gapless divisors, initial A356224 (complement A356225).
%Y A356234 Cf. A000005, A001222, A060680-A060683, A073491-A073495, A193829, A330103, A356233-A356237.
%K A356234 nonn,tabf
%O A356234 1,1
%A A356234 _Gus Wiseman_, Aug 28 2022