A390666 The sum of the unitary 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, 5, 18, 12, 20, 18, 32, 36, 24, 12, 6, 42, 4, 24, 30, 72, 32, 5, 48, 54, 48, 12, 38, 60, 56, 18, 42, 96, 44, 36, 24, 72, 48, 20, 8, 18, 72, 42, 54, 12, 72, 24, 80, 90, 60, 72, 62, 96, 32, 5, 84, 144, 68, 54, 96

Offset

1,2

Links

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

Formula

a(n) = A034448(A019555(n)).

a(n) >= A034448(n) with equality if and only if n is squarefree (A005117).

Multiplicative with a(p^e) = p^(ceiling(e/3)+1) + 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/p^7 + 1/p^8) = 1.0061188236678097400841... .

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. A005117, A019555, A034448, A365481, A390664, A390665.

Cf. A013661, A013663.

Sequence in context: A367503 A048250 A323363 * A392167 A390664 A386471

Adjacent sequences: A390663 A390664 A390665 * A390667 A390668 A390669

Keyword

nonn,mult,easy

Author

Amiram Eldar, Nov 14 2025

Status

approved