A365349 The sum of divisors of the smallest exponentially odd number divisible by n.

1, 3, 4, 15, 6, 12, 8, 15, 40, 18, 12, 60, 14, 24, 24, 63, 18, 120, 20, 90, 32, 36, 24, 60, 156, 42, 40, 120, 30, 72, 32, 63, 48, 54, 48, 600, 38, 60, 56, 90, 42, 96, 44, 180, 240, 72, 48, 252, 400, 468, 72, 210, 54, 120, 72, 120, 80, 90, 60, 360, 62, 96, 320

Offset

1,2

Links

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

Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)

Formula

a(n) = A000203(A356191(n)).

Multiplicative with a(p^e) = (p^(e + 2 - (e mod 2)) - 1)/(p - 1).

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

From Vaclav Kotesovec, Sep 04 2023: (Start)

Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(2*s-2) * zeta(2*s-3) * Product_{p prime} (1 - 1/p^(6*s-7) + 1/p^(5*s-6) + 1/p^(5*s-7) + 1/p^(4*s-4) + 1/p^(4*s-5) - 1/p^(4*s-6) - 1/p^(3*s-3) - 1/p^(3*s-4) - 1/p^(2*s-2)).

Let f(s) = Product_{p prime} (1 - 1/p^(6*s-7) + 1/p^(5*s-6) + 1/p^(5*s-7) + 1/p^(4*s-4) + 1/p^(4*s-5) - 1/p^(4*s-6) - 1/p^(3*s-3) - 1/p^(3*s-4) - 1/p^(2*s-2)), then

Sum_{k=1..n} a(k) ~ n^2 * Pi^4 * f(2) / 144 * (log(n) + 3*gamma - 1/2 + 18*zeta'(2)/Pi^2 + f'(2)/f(2)), where

f(2) = Product_{p prime} (1 - 1/p^2) * (1 - 2/p^2 + 1/p^3) = 6*A065464/Pi^2 = 0.26034448085669554670553581687050222309091096557569931376863612821007515...,

f'(2) = f(2) * Sum_{p prime} 3*(3*p-2) * log(p) / (p^3 - 2*p + 1) = f(2) * 4.40861022247384449961018198035049309399000439627743168713608947117149645... and gamma is the Euler-Mascheroni constant A001620. (End)

Mathematica

f[p_, e_] := (p^(e + 2 - Mod[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]^(f[i, 2] + 2 - f[i, 2]%2) - 1)/(f[i, 1] - 1)); }
(PARI) for(n=1, 100, print1(direuler(p=2, n, 1/(1-X) * 1/(1 - p^2*X^2) * (1 + p*X + p^3*X^2 - p^3*X^3) )[n], ", ")) \\ Vaclav Kotesovec, Sep 04 2023

Crossrefs

Cf. A000203, A356191, A365348.

Sequence in context: A286769 A286115 A130409 * A258468 A327275 A343939

Adjacent sequences: A365346 A365347 A365348 * A365350 A365351 A365352

Keyword

nonn,easy,mult

Author

Amiram Eldar, Sep 02 2023

Status

approved