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

1,2

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).

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)

Mathematica

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

Prog

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

Crossrefs

Cf. A322483, A356191.

Cf. A049599, A282446, A353898.

Sequence in context: A049599 A353898 A366901 * A369015 A369890 A373982

Adjacent sequences: A365548 A365549 A365550 * A365552 A365553 A365554

Keyword

nonn,easy,mult

Author

Amiram Eldar, Sep 08 2023

Status

approved