A365331 The number of divisors of the largest square dividing n.

1, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 3, 3, 1, 3, 3, 1, 1, 1, 5, 1, 1, 1, 9, 1, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 5, 3, 3, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 3, 7, 1, 1, 1, 3, 1, 1, 1, 9, 1, 1, 3, 3, 1, 1, 1, 5, 5, 1, 1, 3, 1, 1, 1

Offset

1,4

Comments

All the terms are odd.

The sum of these divisors is A365332(n).

The number of divisors of the square root of the largest square dividing n is A046951(n).

Links

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

Formula

a(n) = A000005(A008833(n)).

a(n) = 1 if and only if n is squarefree (A005117).

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

Dirichlet g.f.: zeta(s)*zeta(2*s)^2/zeta(4*s).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 5/2.

More precise asymptotics: Sum_{k=1..n} a(k) ~ 5*n/2 + 3*zeta(1/2)*sqrt(n)/Pi^2 * (log(n) + 4*gamma - 2 - 24*zeta'(2)/Pi^2 + zeta'(1/2)/zeta(1/2)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 02 2023

Maple

a:= n-> mul(2*iquo(i[2], 2)+1, i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 01 2023

Mathematica

f[p_, e_] := e + 1 - Mod[e, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = vecprod(apply(x -> x + 1 - x%2, factor(n)[, 2]));
(PARI) a(n) = numdiv(n/core(n)); \\ Michel Marcus, Sep 02 2023

Crossrefs

Cf. A000005, A005117, A008833, A046951, A365332.

Sequence in context: A176187 A372288 A180683 * A214635 A166030 A351149

Adjacent sequences: A365328 A365329 A365330 * A365332 A365333 A365334

Keyword

nonn,easy,mult

Author

Amiram Eldar, Sep 01 2023

Status

approved