A366763 The number of divisors of n that have no exponent 2 in their prime factorization.

1, 2, 2, 2, 2, 4, 2, 3, 2, 4, 2, 4, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 6, 2, 4, 3, 4, 2, 8, 2, 5, 4, 4, 4, 4, 2, 4, 4, 6, 2, 8, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 6, 4, 6, 4, 4, 2, 8, 2, 4, 4, 6, 4, 8, 2, 4, 4, 8, 2, 6, 2, 4, 4, 4, 4, 8, 2, 8, 4, 4, 2, 8, 4, 4, 4

Offset

1,2

Comments

The number of terms of A337050 that divide n.

The sum of these divisors is A366764(n), and the largest of them is A366765(n).

Links

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

Formula

Multiplicative with a(p^e) = max(e, 2);

a(n) <= A000005(n), with equality if and only if n is squarefree (A005117).

a(n) >= A034444(n), with equality if and only if n is cubefree (A004709).

Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s)).

Mathematica

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

Prog

(PARI) a(n) = vecprod(apply(x -> max(x, 2), factor(n)[, 2]));

Crossrefs

Cf. A000005, A004709, A005117, A034444, A337050, A366764, A366765.

Sequence in context: A372380 A322483 A061389 * A138011 A036555 A046927

Adjacent sequences: A366760 A366761 A366762 * A366764 A366765 A366766

Keyword

nonn,easy,mult

Author

Amiram Eldar, Oct 21 2023

Status

approved