A369720 The sum of divisors of the smallest cubefull number that is a multiple of n.

1, 15, 40, 15, 156, 600, 400, 15, 40, 2340, 1464, 600, 2380, 6000, 6240, 31, 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

Links

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

Formula

a(n) = A000203(A356193(n)).

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

a(n) >= A000203(n), with equality if and only if n is cubefull (A036966).

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

Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(3) * zeta(4) * Product_{p prime} (1 - 1/p^3 - 1/p^4 + 1/p^7 + 1/p^12 - 1/p^13) = 1.00015013207437782094... .

Mathematica

f[p_, e_] := (p^If[e <= 2, 4, e + 1]-1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50]

Prog

(PARI) a(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 2] <= 2, f[i, 2] = 3)); sigma(f); }

Crossrefs

Cf. A000203, A036966, A356193, A369717, A369719, A369721.

Cf. A002117, A013662, A183700.

Sequence in context: A126950 A091847 A062222 * A369758 A325659 A223432

Adjacent sequences: A369717 A369718 A369719 * A369721 A369722 A369723

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jan 30 2024

Status

approved