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A295931
Number of ways to write n in the form n = (x^y)^z where x, y, and z are positive integers.
10
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
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}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 29 2017
STATUS
approved