A390665 The sum of the unitary divisors of the smallest cube divisible by n.

1, 9, 28, 9, 126, 252, 344, 9, 28, 1134, 1332, 252, 2198, 3096, 3528, 65, 4914, 252, 6860, 1134, 9632, 11988, 12168, 252, 126, 19782, 28, 3096, 24390, 31752, 29792, 65, 37296, 44226, 43344, 252, 50654, 61740, 61544, 1134, 68922, 86688, 79508, 11988, 3528, 109512

Offset

1,2

Comments

First differs from A351266, A369721 and A369759 at n = 16.

Links

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

Formula

a(n) = A034448(A053149(n)).

a(n) >= A034448(n) with equality if and only if n is a cube (A000578).

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

Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(4) * zeta(9) * Product_{p prime} (1 - 1/p^2 - 1/p^9 + 1/p^10 - 1/p^13 + 1/p^14) = 0.658357292864217429508... .

Mathematica

f[p_, e_] := p^(e + Mod[3 - 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]^(f[i, 2] + (3-f[i, 2])%3) + 1); }

Crossrefs

Cf. A000578, A034448, A053149, A351266, A365479, A369718, A369721, A369759, A390663, A390666.

Cf. A013662, A013667.

Sequence in context: A351266 A369721 A369759 * A033479 A254139 A382945

Adjacent sequences: A390662 A390663 A390664 * A390666 A390667 A390668

Keyword

nonn,mult,easy

Author

Amiram Eldar, Nov 14 2025

Status

approved