A384556 The sum of the exponential divisors of n that are cubefree.

1, 2, 3, 6, 5, 6, 7, 2, 12, 10, 11, 18, 13, 14, 15, 6, 17, 24, 19, 30, 21, 22, 23, 6, 30, 26, 3, 42, 29, 30, 31, 2, 33, 34, 35, 72, 37, 38, 39, 10, 41, 42, 43, 66, 60, 46, 47, 18, 56, 60, 51, 78, 53, 6, 55, 14, 57, 58, 59, 90, 61, 62, 84, 6, 65, 66, 67, 102, 69

Offset

1,2

Comments

The number of these divisors is A056624(n), and the largest of them is A066990(n).

Links

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

Formula

Multiplicative with a(p^e) = p if e is odd, and p+p^2 is e is even.

a(n) <= A051377(n), with equality if and only if n is cubefree (A004709).

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

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4) * Product_{p prime} (1 - 2/p^4 + 1/p^5) = 0.95692470821076622881...

Mathematica

f[p_, e_] := If[OddQ[e], p, p + p^2]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1] + if(!(f[i, 2] % 2), f[i, 1]^2)); }
(Python)
from math import prod
from sympy import factorint
def A384556(n): return prod(p*(1+p*(e&1^1)) for p, e in factorint(n).items()) # Chai Wah Wu, Jun 03 2025

Crossrefs

Cf. A004709, A013662, A051377, A056624, A066990, A073185, A322791, A384554.

Sequence in context: A327669 A253413 A337355 * A392448 A093783 A096861

Adjacent sequences: A384553 A384554 A384555 * A384557 A384558 A384559

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jun 03 2025

Status

approved