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

1,2

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).

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

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 *)

Prog

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

Crossrefs

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

Cf. A019554, A034444, A322483, A365210.

Sequence in context: A365499 A158522 A034444 * A365210 A365488 A365171

Adjacent sequences: A365488 A365489 A365490 * A365492 A365493 A365494

Keyword

nonn,easy,mult

Author

Amiram Eldar, Sep 05 2023

Status

approved