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A296131
Number of twice-factorizations of n where the first factorization is strict and the latter factorizations are constant, i.e., type (P,Q,R).
5
1, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 8, 2, 2, 4, 4, 1, 5, 1, 9, 2, 2, 2, 9, 1, 2, 2, 8, 1, 5, 1, 4, 4, 2, 1, 13, 2, 4, 2, 4, 1, 8, 2, 8, 2, 2, 1, 11, 1, 2, 4, 16, 2, 5, 1, 4, 2, 5, 1, 18, 1, 2, 4, 4, 2, 5, 1, 13, 5, 2, 1, 11, 2
OFFSET
1,4
COMMENTS
a(n) is the number of ways to choose a perfect divisor of each factor in a strict factorization of n.
FORMULA
Dirichlet g.f.: Product_{n > 1}(1 + A089723(n)/n^s).
EXAMPLE
The a(24) = 8 twice-factorizations: (2)*(3)*(2*2), (2)*(3)*(4), (2)*(12), (3)*(2*2*2), (3)*(8), (2*2)*(6), (4)*(6), (24).
MATHEMATICA
sfs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[sfs[n/d], Min@@#>d&]], {d, Rest[Divisors[n]]}]];
Table[Sum[Product[DivisorSigma[0, GCD@@FactorInteger[d][[All, 2]]], {d, fac}], {fac, sfs[n]}], {n, 100}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 05 2017
STATUS
approved