OFFSET
1,2
COMMENTS
a(n) depends only on the prime signature of n. - Andrew Howroyd, Nov 18 2018
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..10000
Jacob Sprittulla, On Colored Factorizations, arXiv:2008.09984 [math.CO], 2020.
FORMULA
Dirichlet g.f.: Product_{n > 1} 1/(1 - n^(-s))^2. [corrected by Ilya Gutkovskiy, Dec 14 2020]
a(p^n) = A000712(n) for prime p. - Andrew Howroyd, Nov 18 2018
EXAMPLE
The a(6) = 6 factorizations: (2*3)*(), (3)*(2), (2)*(3), ()*(2*3), (6)*(), ()*(6).
The a(12) = 16 factorizations:
()*(2*2*3), (2)*(2*3), (3)*(2*2), (2*2)*(3), (2*3)*(2), (2*2*3)*(),
()*(2*6), (2)*(6), (6)*(2), (2*6)*(), ()*(3*4), (3)*(4), (4)*(3), (3*4)*(),
()*(12), (12)*().
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Sum[Length[facs[d]]*Length[facs[n/d]], {d, Divisors[n]}], {n, 100}]
PROG
(PARI) MultEulerT(u)={my(v=vector(#u)); v[1]=1; for(k=2, #u, forstep(j=#v\k*k, k, -k, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j]+=binomial(e+u[k]-1, e)*v[i]))); v}
seq(n)={MultEulerT(vector(n, i, 2))} \\ Andrew Howroyd, Nov 18 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 27 2018
STATUS
approved