A365551 The number of exponentially odd divisors of the smallest exponentially odd number divisible by n.

1, 2, 2, 3, 2, 4, 2, 3, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 3, 6, 2, 8, 2, 4, 4, 4, 4, 9, 2, 4, 4, 6, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 6, 4, 6, 4, 4, 2, 12, 2, 4, 6, 5, 4, 8, 2, 6, 4, 8, 2, 9, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4, 4

Offset: 1

Author

Amiram Eldar, Sep 08 2023

Keywords

• nonn
• easy
• mult

Comments

First differs from A049599 and A282446 at n = 32, and from A353898 at n = 64.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms).

Crossrefs

Cf. A322483, A356191.

Cf. A049599, A282446, A353898.

Programs

Mathematica

f[p_, e_] := Ceiling[(e + 3)/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = vecprod(apply(x -> ceil((x+3)/2), factor(n)[, 2]));

Formula

a(n) = A322483(A356191(n)).

Multiplicative with a(p^e) = ceiling((e+3)/2).

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

From Vaclav Kotesovec, Sep 09 2023: (Start)

Let f(s) = Product_{p prime} (1 - 1/p^(2*s) - 1/p^(3*s) + 1/p^(4*s)).

Dirichlet g.f.: zeta(s)^2 * zeta(2*s) * f(s).

Sum_{k=1..n} a(k) ~ (Pi^2 * f(1) * n / 6) * (log(n) + 2*gamma - 1 + 12*zeta'(2)/Pi^2 + f'(1)/f(1)), where

f(1) = Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = 0.5358961538283379998085026313185459506482223745141452711510108346133288119...,

f'(1) = f(1) * Sum_{p prime} (-4 + 3*p + 2*p^2) * log(p) / (1 - p - p^2 + p^4) = f(1) * 1.452592479445159559037143959382854734148246511441192913672347667991...

and gamma is the Euler-Mascheroni constant A001620. (End)