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A316439 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).

%I A316439 #11 Aug 11 2019 19:41:20
%S A316439 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 A316439 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 A316439 1,1,1,1,4,3,1,1,1,1,1,1,1,3,2,1,1,1,3
%N A316439 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 A316439 Alois P. Heinz, <a href="/A316439/b316439.txt">Rows n = 1..20000, flattened</a>
%e A316439 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 A316439 Triangle begins:
%e A316439   {}
%e A316439   1
%e A316439   1
%e A316439   1  1
%e A316439   1
%e A316439   1  1
%e A316439   1
%e A316439   1  1  1
%e A316439   1  1
%e A316439   1  1
%e A316439   1
%e A316439   1  2  1
%e A316439   1
%e A316439   1  1
%e A316439   1  1
%e A316439   1  2  1  1
%e A316439   1
%e A316439   1  2  1
%e A316439   1
%e A316439   1  2  1
%e A316439   1  1
%e A316439   1  1
%e A316439   1
%e A316439   1  3  2  1
%e A316439   1  1
%e A316439   1  1
%e A316439   1  1  1
%e A316439   1  2  1
%e A316439   1
%e A316439   1  3  1
%p A316439 g:= proc(n, k) option remember; `if`(n>k, 0, x)+
%p A316439       `if`(isprime(n), 0, expand(x*add(`if`(d>k, 0,
%p A316439       g(n/d, d)), d=numtheory[divisors](n) minus {1, n})))
%p A316439     end:
%p A316439 T:= n-> `if`(n=1, [][], (p-> seq(coeff(p, x, i)
%p A316439         , i=1..degree(p)))(g(n$2))):
%p A316439 seq(T(n), n=1..50);  # _Alois P. Heinz_, Aug 11 2019
%t A316439 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A316439 Table[Length[Select[facs[n],Length[#]==k&]],{n,100},{k,PrimeOmega[n]}]
%Y A316439 Cf. A001222 (row lengths), A001055 (row sums), A001970, A007716, A045778, A162247, A259936, A281116, A303386.
%K A316439 nonn,tabf
%O A316439 1,18
%A A316439 _Gus Wiseman_, Jul 03 2018