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.

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

Author

Gus Wiseman, Mar 30 2017

Keywords

• nonn

Comments

a(1)=infinity is not an integer so is necessarily omitted from the data.

Examples

			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.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 2..1000

Crossrefs

Cf. A001055, A001597, A007916, A089723.

Programs

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}]