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

1,4

Comments

The number of these divisors is A365333(n).

Links

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

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)).

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]

Prog

(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); }

Crossrefs

Cf. A008833, A005117, A033634, A365333.

Sequence in context: A365336 A365404 A069290 * A348660 A076476 A243200

Adjacent sequences: A365331 A365332 A365333 * A365335 A365336 A365337

Keyword

nonn,easy,mult

Author

Amiram Eldar, Sep 01 2023

Status

approved