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A296132
Number of twice-factorizations of n where the first factorization is constant and the latter factorizations are strict, i.e., type (P,R,Q).
6
1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 5, 2, 2, 3, 3, 1, 5, 1, 4, 2, 2, 2, 9, 1, 2, 2, 5, 1, 5, 1, 3, 3, 2, 1, 7, 2, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 9, 1, 2, 3, 10, 2, 5, 1, 3, 2, 5, 1, 9, 1, 2, 3, 3, 2, 5, 1, 7, 4, 2, 1, 9, 2, 2, 2
OFFSET
1,4
COMMENTS
a(n) is also the number of ways to choose a perfect divisor d|n and then a sequence of log_d(n) strict factorizations of d.
EXAMPLE
The a(36) = 9 twice-factorizations are (2*3)*(2*3), (2*3)*(6), (6)*(2*3), (6)*(6), (2*3*6), (2*18), (3*12), (4*9), (36).
MATHEMATICA
sfs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[sfs[n/d], Min@@#>d&]], {d, Rest[Divisors[n]]}]];
Table[Sum[Length[sfs[n^(1/g)]]^g, {g, Divisors[GCD@@FactorInteger[n][[All, 2]]]}], {n, 100}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 05 2017
STATUS
approved