A369758 The sum of divisors of the smallest cubefull exponentially odd number that is divisible by n.

1, 15, 40, 15, 156, 600, 400, 15, 40, 2340, 1464, 600, 2380, 6000, 6240, 63, 5220, 600, 7240, 2340, 16000, 21960, 12720, 600, 156, 35700, 40, 6000, 25260, 93600, 30784, 63, 58560, 78300, 62400, 600, 52060, 108600, 95200, 2340, 70644, 240000, 81400, 21960, 6240

Offset

1,2

Comments

First differs from A369720 at n = 16.

Links

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

Formula

a(n) = A000203(A356192(n)).

Multiplicative with a(p) = p^3 + p^2 + p + 1, a(p^e) = (p^(e+1)-1)/(p-1) for an odd e >= 3, and a(p^e) = (p^(e+2)-1)/(p-1) for an even e.

a(n) >= A000203(n), with equality if and only if n is cubefull exponentially odd number (A335988).

Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-3) + 1/p^(s-2) + 1/p^(s-1) - 1/p^(2*s-2) - 1/p^(3*s-5) - 1/p^(3*s-4) - 1/p^(3*s-3) + 1/p^(4*s-5) + 1/p^(4*s-4)).

Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(4) * zeta(6) * Product_{p prime} (1 - 1/p^4 - 1/p^6 + 1/p^10 + 1/p^11 - 1/p^13) = 1.00040193512214077945... .

Equivalently, c = Product_{p prime} (1 + 1/(p^3*(p^4 - 1)*(p^4 + p^2 + 1))). - Vaclav Kotesovec, Feb 02 2024

Mathematica

f[p_, e_] := (p^If[OddQ[e], Max[e, 3] + 1, e + 2] - 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]^if(f[i, 2]%2, max(f[i, 2], 3) + 1, f[i, 2] + 2) - 1)/(f[i, 1] - 1)); }
(Python)
from math import prod
from sympy import factorint
def A369758(n): return prod((p**((3 if e==1 else e)+1+(e&1^1))-1)//(p-1) for p, e in factorint(n).items()) # Chai Wah Wu, Feb 03 2024

Crossrefs

Cf. A000203, A335988, A356192, A365349, A369720, A369757, A369759.

Cf. A013662, A013664.

Sequence in context: A091847 A062222 A369720 * A325659 A223432 A044092

Adjacent sequences: A369755 A369756 A369757 * A369759 A369760 A369761

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jan 31 2024

Status

approved