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

Author

Amiram Eldar, Sep 01 2023

Keywords

• nonn
• easy
• mult

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

Crossrefs

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

Programs

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]

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

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