A365334 The sum of exponentially odd divisors of the largest square dividing n.

1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1, 1, 11, 1, 4, 1, 3, 1, 1, 1, 3, 6, 1, 4, 3, 1, 1, 1, 11, 1, 1, 1, 12, 1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 11, 8, 6, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 4, 43, 1, 1, 1, 3, 1, 1, 1, 12, 1, 1, 6, 3, 1, 1, 1, 11, 31, 1, 1, 3, 1

Offset: 1

Author

Amiram Eldar, Sep 01 2023

Keywords

• nonn
• easy
• mult

Comments

The number of these divisors is A365333(n).

Links

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

Crossrefs

Cf. A008833, A005117, A033634, A365333.

Programs

Mathematica

f[p_, e_] := (p^(e + 1 - Mod[e, 2]) - p)/(p^2 - 1) + 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) - f[i,1])/(f[i,1]^2 - 1) + 1);}

Formula

a(n) = A033634(A008833(n)).

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

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

Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 - 1/p^(2*s-2) + 1/p^(2*s-1)).