A392086 The unitary 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, 3, 16, 2, 18, 4, 12, 10, 22, 2, 4, 12, 2, 6, 28, 8, 30, 3, 20, 16, 24, 2, 36, 18, 24, 4, 40, 12, 42, 10, 8, 22, 46, 6, 6, 4, 32, 12, 52, 2, 40, 6, 36, 28, 58, 8, 60, 30, 12, 3, 48, 20, 66, 16, 44, 24, 70, 2, 72, 36, 8, 18, 60

Offset

1,3

Links

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

Formula

a(n) = A047994(A019555(n)).

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

Dirichlet g.f.: zeta(s) * zeta(3*s-1) * Product_{p prime} (1 + 1/p^(s-1) - 2/p^s - 1/p^(3*s-1) + 1/p^(4*s-1)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(2) * zeta(5) * Product_{p prime} (1 - 3/p^2 + 2/p^3 - 1/p^5 + 1/p^6 + 1/p^7 - 1/p^8) = 0.47590270384210734089... .

Mathematica

f[p_, e_] := p^Ceiling[e/3] - 1; 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]^ceil(f[i, 2]/3)-1); }

Crossrefs

Cf. A013661, A013663, A047994, A019555, A390753.

Sequence in context: A023900 A141564 A390753 * A374840 A239641 A249151

Adjacent sequences: A392083 A392084 A392085 * A392087 A392088 A392089

Keyword

nonn,mult,easy

Author

Amiram Eldar, Dec 30 2025

Status

approved