login
A284639
Number of ways to write n > 1 as a power of a product n = (x_1*..*x_k)^y, where y >= 1, x_i > 1, and x_i <= x_j for i < j.
16
1, 1, 3, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 2, 8, 1, 4, 1, 4, 2, 2, 1, 7, 3, 2, 4, 4, 1, 5, 1, 8, 2, 2, 2, 11, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 3, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 17, 2, 5, 1, 4, 2, 5, 1, 16, 1, 2, 4, 4, 2, 5, 1, 12, 8, 2, 1, 11, 2
OFFSET
2,3
COMMENTS
a(1)=infinity is not an integer so is necessarily omitted from the data.
LINKS
EXAMPLE
The a(64)=17 ways to write 64 as a power of a product:
(2*2*2*2*2*2)^1 (2*2*2)^2 (2*2)^3 (2)^6
(2*2*2*2*4)^1 (2*4)^2 (4)^3
(2*2*2*8)^1 (8)^2
(2*2*4*4)^1
(2*2*16)^1
(2*4*8)^1
(2*32)^1
(4*4*4)^1
(4*16)^1
(8*8)^1
(64)^1.
MATHEMATICA
nn=85;
postfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[postfacs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[DivisorSum[GCD@@FactorInteger[n][[All, 2]], Length[postfacs[n^(1/#)]]&], {n, 2, nn}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 30 2017
STATUS
approved