A284639 - OEIS

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

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OFFSET

2,3

COMMENTS

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

LINKS

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

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

Cf. A001055, A001597, A007916, A089723.

Sequence in context: A203531 A324885 A046645 * A320887 A295923 A325806

Adjacent sequences: A284636 A284637 A284638 * A284640 A284641 A284642

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 30 2017

STATUS

approved