A390664 The sum of the divisors of the smallest number whose cube is divisible by n.

1, 3, 4, 3, 6, 12, 8, 3, 4, 18, 12, 12, 14, 24, 24, 7, 18, 12, 20, 18, 32, 36, 24, 12, 6, 42, 4, 24, 30, 72, 32, 7, 48, 54, 48, 12, 38, 60, 56, 18, 42, 96, 44, 36, 24, 72, 48, 28, 8, 18, 72, 42, 54, 12, 72, 24, 80, 90, 60, 72, 62, 96, 32, 7, 84, 144, 68, 54, 96

Offset

1,2

Links

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

Formula

a(n) = A000203(A019555(n)).

a(n) <= A000203(n) with equality if and only if n is squarefree (A005117).

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

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(2) * zeta(5) * Product_{p prime} (1 - 1/p^2 - 1/p^5 + 1/p^6) = 1.01185914843439887476... .

Mathematica

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

Crossrefs

Cf. A000203, A005117, A019555, A365347, A365488, A390663.

Cf. A013661, A013663.

Sequence in context: A323363 A390666 A392167 * A386471 A073181 A183100

Adjacent sequences: A390661 A390662 A390663 * A390665 A390666 A390667

Keyword

nonn,mult,easy

Author

Amiram Eldar, Nov 14 2025

Status

approved