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Irregular triangle where T(n,k) is the number of factorizations of n into k factors > 1, with k ranging from 1 to Omega(n).
32

%I #11 Aug 11 2019 19:41:20

%S 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,2,1,1,1,

%T 2,1,1,1,1,1,1,1,3,2,1,1,1,1,1,1,1,1,1,2,1,1,1,3,1,1,1,2,2,1,1,1,1,1,

%U 1,1,1,1,4,3,1,1,1,1,1,1,1,3,2,1,1,1,3

%N Irregular triangle where T(n,k) is the number of factorizations of n into k factors > 1, with k ranging from 1 to Omega(n).

%H Alois P. Heinz, <a href="/A316439/b316439.txt">Rows n = 1..20000, flattened</a>

%e The factorizations of 24 are (2*2*2*3), (2*2*6), (2*3*4), (2*12), (3*8), (4*6), (24) so the 24th row is {1, 3, 2, 1}.

%e Triangle begins:

%e {}

%e 1

%e 1

%e 1 1

%e 1

%e 1 1

%e 1

%e 1 1 1

%e 1 1

%e 1 1

%e 1

%e 1 2 1

%e 1

%e 1 1

%e 1 1

%e 1 2 1 1

%e 1

%e 1 2 1

%e 1

%e 1 2 1

%e 1 1

%e 1 1

%e 1

%e 1 3 2 1

%e 1 1

%e 1 1

%e 1 1 1

%e 1 2 1

%e 1

%e 1 3 1

%p g:= proc(n, k) option remember; `if`(n>k, 0, x)+

%p `if`(isprime(n), 0, expand(x*add(`if`(d>k, 0,

%p g(n/d, d)), d=numtheory[divisors](n) minus {1, n})))

%p end:

%p T:= n-> `if`(n=1, [][], (p-> seq(coeff(p, x, i)

%p , i=1..degree(p)))(g(n$2))):

%p seq(T(n), n=1..50); # _Alois P. Heinz_, Aug 11 2019

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t Table[Length[Select[facs[n],Length[#]==k&]],{n,100},{k,PrimeOmega[n]}]

%Y Cf. A001222 (row lengths), A001055 (row sums), A001970, A007716, A045778, A162247, A259936, A281116, A303386.

%K nonn,tabf

%O 1,18

%A _Gus Wiseman_, Jul 03 2018