A295931 Number of ways to write n in the form n = (x^y)^z where x, y, and z are positive integers.

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, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1

Offset

1,4

Comments

By convention a(1) = 1.

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

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

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

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

Example

The a(256) = 10 ways are:
(2^1)^8    (2^2)^4   (2^4)^2  (2^8)^1
(4^1)^4    (4^2)^2   (4^4)^1
(16^1)^2   (16^2)^1
(256^1)^1

Maple

f:= proc(n) local m, d, t;
  m:= igcd(seq(t[2], t=ifactors(n)[2]));
  add(numtheory:-tau(d), d=numtheory:-divisors(m))
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Dec 19 2017

Mathematica

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

Crossrefs

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

Sequence in context: A030401 A275888 A308166 * A295920 A176187 A180683

Adjacent sequences: A295928 A295929 A295930 * A295932 A295933 A295934

Keyword

nonn

Author

Gus Wiseman, Nov 29 2017

Status

approved