A390753 The Euler totient of the smallest number whose cube is divisible by n.

1, 1, 2, 1, 4, 2, 6, 1, 2, 4, 10, 2, 12, 6, 8, 2, 16, 2, 18, 4, 12, 10, 22, 2, 4, 12, 2, 6, 28, 8, 30, 2, 20, 16, 24, 2, 36, 18, 24, 4, 40, 12, 42, 10, 8, 22, 46, 4, 6, 4, 32, 12, 52, 2, 40, 6, 36, 28, 58, 8, 60, 30, 12, 2, 48, 20, 66, 16, 44, 24, 70, 2, 72, 36

Offset

1,3

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = A000010(A019555(n)).

Multiplicative with a(p^e) = (p-1) * p^floor((e-1)/3).

Dirichlet g.f.: zeta(3*s-1) * Product_{p prime} (1 + 1/p^(s-1) - 1/p^s + 1/p^(2*s-1) - 1/p^(2*s) - 1/p^(3*s)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(5) * Product_{p prime} (1 - 2/p^2 + 2/p^3 - 2/p^4 + 1/p^5 - 1/p^6 + 1/p^7) = 0.47386381252034025417... .

Mathematica

f[p_, e_] := (p-1) * p^Floor[(e-1)/3]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]-1) * f[i, 1]^((f[i, 2]-1)\3)); }

Crossrefs

Cf. A000010, A013663, A019555, A390752.

Sequence in context: A173557 A023900 A141564 * A392086 A374840 A239641

Adjacent sequences: A390750 A390751 A390752 * A390754 A390755 A390756

Keyword

nonn,mult,easy

Author

Amiram Eldar, Nov 17 2025

Status

approved