cp's OEIS Frontend

A296119 Number of ways to choose a strict factorization of each factor in a factorization of n.

%I A296119 #11 Oct 08 2018 08:08:31
%S A296119 1,1,1,2,1,3,1,4,2,3,1,7,1,3,3,7,1,7,1,7,3,3,1,16,2,3,4,7,1,12,1,12,3,
%T A296119 3,3,21,1,3,3,16,1,12,1,7,7,3,1,33,2,7,3,7,1,16,3,16,3,3,1,34,1,3,7,
%U A296119 23,3,12,1,7,3,12,1,50,1,3,7,7,3,12,1,33,7,3
%N A296119 Number of ways to choose a strict factorization of each factor in a factorization of n.
%H A296119 Antti Karttunen, <a href="/A296119/b296119.txt">Table of n, a(n) for n = 1..16384</a>
%H A296119 Antti Karttunen, <a href="/A296119/a296119.txt">Data supplement: n, a(n) computed for n = 1..100000</a>
%F A296119 Dirichlet g.f.: 1/Product_{n > 1}(1 - A045778(n)/n^s).
%e A296119 The a(24) = 16 twice-factorizations:
%e A296119 (2)*(2)*(2)*(3),
%e A296119 (2)*(2)*(2*3), (2)*(2)*(6), (2)*(3)*(4),
%e A296119 (2)*(2*6), (2)*(3*4), (2)*(12), (3)*(2*4), (3)*(8), (4)*(2*3), (4)*(6),
%e A296119 (2*3*4), (2*12), (3*8), (4*6), (24).
%t A296119 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A296119 Table[Sum[Times@@(Length[Select[facs[#],UnsameQ@@#&]]&/@fac),{fac,facs[n]}],{n,100}]
%o A296119 (PARI)
%o A296119 A045778(n, m=n) = ((n<=m) + sumdiv(n, d, if((d>1)&&(d<=m)&&(d<n), A045778(n/d, d-1))));
%o A296119 A296119(n, m=n) = if(1==n, 1, sumdiv(n, d, if((d>1)&&(d<=m), A045778(d)*A296119(n/d, d)))); \\ _Antti Karttunen_, Oct 08 2018
%Y A296119 Cf. A000009, A005117, A045778, A270995, A281113, A294788, A296118, A296120, A296121.
%K A296119 nonn
%O A296119 1,4
%A A296119 _Gus Wiseman_, Dec 05 2017