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A301830
Number of factorizations of n into factors (greater than 1) of two kinds.
19
1, 2, 2, 5, 2, 6, 2, 10, 5, 6, 2, 16, 2, 6, 6, 20, 2, 16, 2, 16, 6, 6, 2, 36, 5, 6, 10, 16, 2, 22, 2, 36, 6, 6, 6, 46, 2, 6, 6, 36, 2, 22, 2, 16, 16, 6, 2, 76, 5, 16, 6, 16, 2, 36, 6, 36, 6, 6, 2, 64, 2, 6, 16, 65, 6, 22, 2, 16, 6, 22, 2, 108, 2, 6, 16, 16, 6
OFFSET
1,2
COMMENTS
a(n) depends only on the prime signature of n. - Andrew Howroyd, Nov 18 2018
LINKS
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 27 2018
STATUS
approved