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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

%I

%S 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,

%T 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,

%U 4,2,5,1,16,1,2,4,4,2,5,1,12,8,2,1,11,2

%N 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.

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

%H Vincenzo Librandi, <a href="/A284639/b284639.txt">Table of n, a(n) for n = 2..1000</a>

%e The a(64)=17 ways to write 64 as a power of a product:

%e (2*2*2*2*2*2)^1 (2*2*2)^2 (2*2)^3 (2)^6

%e (2*2*2*2*4)^1 (2*4)^2 (4)^3

%e (2*2*2*8)^1 (8)^2

%e (2*2*4*4)^1

%e (2*2*16)^1

%e (2*4*8)^1

%e (2*32)^1

%e (4*4*4)^1

%e (4*16)^1

%e (8*8)^1

%e (64)^1.

%t nn=85;

%t postfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[postfacs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t Table[DivisorSum[GCD@@FactorInteger[n][[All,2]],Length[postfacs[n^(1/#)]]&],{n,2,nn}]

%Y Cf. A001055, A001597, A007916, A089723.

%K nonn

%O 2,3

%A _Gus Wiseman_, Mar 30 2017

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Last modified August 7 11:08 EDT 2020. Contains 336275 sequences. (Running on oeis4.)