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Irregular triangle read by rows where row n is the ordered factorization of n into maximal gapless divisors.
11

%I #5 Aug 30 2022 09:41:31

%S 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 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 43,4,11,45,2,23,47,48,49,2,25,3,17,4,13,53,54,5,11,8

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

%C Row-products are the positive integers 1, 2, 3, ...

%e The first 16 rows:

%e 1 =

%e 2 = 2

%e 3 = 3

%e 4 = 4

%e 5 = 5

%e 6 = 6

%e 7 = 7

%e 8 = 8

%e 9 = 9

%e 10 = 2 * 5

%e 11 = 11

%e 12 = 12

%e 13 = 13

%e 14 = 2 * 7

%e 15 = 15

%e 16 = 16

%e The factorization of 18564 is 18564 = 12*7*221, so row 18564 is {12,7,221}.

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Table[Times@@Prime/@#&/@Split[primeMS[n],#1>=#2-1&],{n,100}]

%Y Row-lengths are A287170, firsts A066205, even bisection A356229.

%Y Applying bigomega to all parts gives A356226, statistics A356227-A356232.

%Y A001055 counts factorizations.

%Y A001221 counts distinct prime factors, sum A001414.

%Y A003963 multiplies together the prime indices.

%Y A056239 adds up the prime indices, row sums of A112798.

%Y A132747 counts non-isolated divisors, complement A132881.

%Y A356069 counts gapless divisors, initial A356224 (complement A356225).

%Y Cf. A000005, A001222, A060680-A060683, A073491-A073495, A193829, A330103, A356233-A356237.

%K nonn,tabf

%O 1,1

%A _Gus Wiseman_, Aug 28 2022