A365332 -id:A365332 - cp's OEIS frontend

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

Amiram Eldar, Sep 01 2023

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).

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

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

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

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

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

a(n) = numdiv(n/core(n)); \\ Michel Marcus, Sep 02 2023

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

A380162 a(n) is the value of the Euler totient function when applied to the largest square dividing n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 6, 1, 1, 2, 1, 1, 1, 8, 1, 6, 1, 2, 1, 1, 1, 2, 20, 1, 6, 2, 1, 1, 1, 8, 1, 1, 1, 12, 1, 1, 1, 2, 1, 1, 1, 2, 6, 1, 1, 8, 42, 20, 1, 2, 1, 6, 1, 2, 1, 1, 1, 2, 1, 1, 6, 32, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 20, 2, 1, 1, 1, 8, 54, 1, 1, 2, 1

Offset: 1

Amiram Eldar, Jan 13 2025

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

Cf. A000010, A000290, A005117, A008833, A077591, A365331, A365332, A380163.

Cf. A002117, A013661, A078434.

f[p_, e_] := If[e == 1, 1, (p-1)*p^(2*Floor[e/2]-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, 1, (f[i, 1]-1) * f[i, 1]^(2*(f[i, 2]\2)-1)));}

a(n) = A000010(A008833(n)).
a(n) >= 1, with equality if and only if n is squarefree (A005117).
a(n) <= A000010(n), with equality if and only if n is either a square (A000290) or twice an odd square (A077591 \ {1}).
Multiplicative with a(p) = 1, and a(p^e) = (p-1)*p^(2*floor(e/2)-1) if e >= 2.
Dirichlet g.f.: zeta(s) * zeta(2*s-2) / (zeta(2*s-1) * zeta(2*s)).
Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = zeta(3/2)/(zeta(2)*zeta(3)) = 1.32118019580177760682... .

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

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A380162 a(n) is the value of the Euler totient function when applied to the largest square dividing n.

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