A363332 a(n) is the number of divisors of n that are both coreful and bi-unitary.

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Amiram Eldar, May 28 2023

Keywords

• nonn
• easy
• mult

Comments

For the definition of a coreful divisor see A307958, and for the definition of a bi-unitary divisor see A222266.

If e > 0 is the exponent of the highest power of p dividing n (where p is a prime), then for each divisor d of n that is both a coreful and an bi-unitary divisor, the exponent of the highest power of p dividing d is a number k >= 1 that is not equal to e/2.

All the terms are odd.

Examples

			a(8) = 3 since 8 has 4 divisors, 1, 2, 4 and 8, all are bi-unitary and 3 of them (2, 4 and 8) are also coreful.
		

Links

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

Crossrefs

Cf. A004709, A005361 (number of coreful divisors), A222266, A286324, A362852.

Programs

Mathematica

f[p_, e_] := If[OddQ[e], e, e - 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 120]

PARI

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

Formula

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

a(n) = 1 if and only if n is cubefree (A004709).

a(n) >= A362852(n), with equality if and only if n is cubefree.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 2/(p^3-p)) = 1.48264570900305853294... .

Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 - 1/p^(2*s) + 2/p^(3*s)). - Amiram Eldar, Sep 24 2023