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Number of ways to write n in the form n = (x^y)^z where x, y, and z are positive integers.
10

%I #20 Dec 19 2017 16:08:24

%S 1,1,1,3,1,1,1,3,3,1,1,1,1,1,1,6,1,1,1,1,1,1,1,1,3,1,3,1,1,1,1,3,1,1,

%T 1,3,1,1,1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,9,1,1,1,1,

%U 1,1,1,1,1,1,1,1,1,1,1,1,6,1,1,1,1,1,1

%N Number of ways to write n in the form n = (x^y)^z where x, y, and z are positive integers.

%C By convention a(1) = 1.

%C Values can be 1, 3, 6, 9, 10, 15, 18, 21, 27, 28, 30, 36, 45, 54, 60, 63, 84, 90, etc. - _Robert G. Wilson v_, Dec 10 2017

%H Robert Israel, <a href="/A295931/b295931.txt">Table of n, a(n) for n = 1..10000</a>

%F a(A175082(k)) = 1, a(A093771(k)) = 3.

%F a(n) = Sum_{d|A052409(n)} A000005(d).

%e The a(256) = 10 ways are:

%e (2^1)^8 (2^2)^4 (2^4)^2 (2^8)^1

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

%e (16^1)^2 (16^2)^1

%e (256^1)^1

%p f:= proc(n) local m,d,t;

%p m:= igcd(seq(t[2],t=ifactors(n)[2]));

%p add(numtheory:-tau(d),d=numtheory:-divisors(m))

%p end proc:

%p f(1):= 1:

%p map(f, [$1..100]); # _Robert Israel_, Dec 19 2017

%t Table[Sum[DivisorSigma[0,d],{d,Divisors[GCD@@FactorInteger[n][[All,2]]]}],{n,100}]

%Y Cf. A000005, A007425, A052409, A052410, A089723, A093771, A175082, A277562, A281113, A284639, A294786, A294336, A294338, A295920, A295923, A295924, A295935.

%K nonn

%O 1,4

%A _Gus Wiseman_, Nov 29 2017