A365332 The sum of divisors of the largest square dividing n.

1, 1, 1, 7, 1, 1, 1, 7, 13, 1, 1, 7, 1, 1, 1, 31, 1, 13, 1, 7, 1, 1, 1, 7, 31, 1, 13, 7, 1, 1, 1, 31, 1, 1, 1, 91, 1, 1, 1, 7, 1, 1, 1, 7, 13, 1, 1, 31, 57, 31, 1, 7, 1, 13, 1, 7, 1, 1, 1, 7, 1, 1, 13, 127, 1, 1, 1, 7, 1, 1, 1, 91, 1, 1, 31, 7, 1, 1, 1, 31, 121

Offset: 1

Author

Amiram Eldar, Sep 01 2023

Keywords

• nonn
• easy
• mult

Comments

All the terms are odd.

The number of these divisors is A365331(n).

The sum of divisors of the square root of the largest square dividing n is A069290(n).

Links

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

Crossrefs

Cf. A000203, A005117, A008833, A069290, A078434, A365331.

Programs

Mathematica

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

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^(f[i,2] + 1 - f[i,2]%2) - 1)/(f[i,1] - 1));}

Formula

a(n) = A000203(A008833(n)).

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

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

Dirichlet g.f.: zeta(s) * zeta(2*s-1) * zeta(2*s-2) / zeta(4*s-2).

Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = 5*zeta(3/2)/Pi^2 = 1.323444812234... .