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

Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

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)).

From Vaclav Kotesovec, Apr 20 2025: (Start)

Let f(s) = Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s)).

Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where

f(1) = Product_{p prime} (1 - 1/p^2 + 1/p^3) = A330596 = 0.74853525968236356464421504863791060164164034300532440451585279392592558689...,

f'(1) = f(1) * Sum_{p prime} (2*p-3)*log(p)/(p^3-p+1) = f(1) * 0.560697508735949606541137451100554649565120075155278833396722097786365686597...

and gamma is the Euler-Mascheroni constant A001620. (End)

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.

Cf. A330596.

Sequence in context: A380922 A322483 A061389 * A138011 A036555 A396687

Adjacent sequences: A366760 A366761 A366762 * A366764 A366765 A366766

Keyword

nonn,easy,mult

Author

Amiram Eldar, Oct 21 2023

Status

approved