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A380397
The number of nonunitary divisors of n that are cubes.
2
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0
OFFSET
1
COMMENTS
First differs from A295884 at n = 128.
LINKS
FORMULA
a(n) = Sum_{d|n, gcd(d, n/d) > 1} [d is cube], where [] is the Iverson bracket.
a(n) = A061704(n) - A380395(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(3)*(1 - 1/zeta(4)) = 0.091430367833446168224... .
EXAMPLE
a(16) = 1 since 16 has 1 nonunitary divisor that is a cube, 8 = 2^3.
a(128) = 2 since 128 has 2 nonunitary divisors that are cubes, 8 = 2^3 and 64 = 4^3.
MATHEMATICA
f1[p_, e_] := 1 + Floor[e/3]; f2[p_, e_] := 2^If[Divisible[e, 3], 1, 0]; a[1] = 0; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n])- Times @@ f2 @@@ fct; Array[a, 100]
PROG
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 2]\3) - prod(i = 1, #f~, 2^if(f[i, 2]%3, 0, 1)); }
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Jan 23 2025
STATUS
approved