A365491 The number of divisors of the smallest number whose 4th power is divisible by n.

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 4, 2, 4, 2, 4, 2, 4, 2, 8, 2, 3, 4, 4, 4, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 4, 4, 4, 4, 4, 2, 8, 2, 4, 4, 3, 4, 8, 2, 4, 4, 8, 2, 4, 2, 4, 4, 4, 4, 8, 2, 4, 2, 4, 2, 8, 4, 4, 4

Offset: 1

Author

Amiram Eldar, Sep 05 2023

Keywords

• nonn
• easy
• mult

Comments

First differs from A365210 at n = 25 and from A034444 at n = 32.

The number of divisors of the smallest 4th divisible by n, A053167(n), is A365492(n).

Links

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

Crossrefs

Cf. A000005, A001620, A005117, A053166, A053167, A365492, A365499.

Cf. A019554, A034444, A322483, A365210.

Programs

Mathematica

f[p_, e_] := Ceiling[e/4] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
With[{c=Range[100]^4},Table[DivisorSigma[0,Surd[SelectFirst[c,Mod[#,n]==0&],4]],{n,90}]] (* Harvey P. Dale, Jul 09 2024 *)

PARI

a(n) = vecprod(apply(x -> (x-1)\4 + 2, factor(n)[, 2]));

Formula

a(n) = A000005(A053166(n)).

Multiplicative with a(p^e) = ceiling(e/4) + 1.

a(n) <= A000005(n) with equality if and only if n is squarefree (A005117).

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

From Vaclav Kotesovec, Sep 06 2023: (Start)

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

Let f(s) = Product_{p prime} (1 - 1/p^(2*s) - 1/p^(4*s) + 1/p^(5*s)).

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

f(1) = Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 0.57615273538566705952061107826411727540624711680289618854325028459572487...,

f'(1) = f(1) * Sum_{p prime} (-5 + 4*p + 2*p^3) * log(p) / (1 - p - p^3 + p^5) = f(1) * 1.3011434396559802378314782600747661399223385669839998680418996210...

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

a(n) = A322483(A019554(n)) (the number of exponentially odd divisors of the smallest number whose square is divisible by n). - Amiram Eldar, Sep 08 2023