A368105 The number of bi-unitary divisors of n that are powerful (A001694).

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

Offset: 1

Author

Amiram Eldar, Dec 12 2023

Keywords

• nonn
• easy
• mult

Comments

First differs from A095691 and A365552 at n = 32.

Links

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

Crossrefs

Cf. A013661, A001694, A005117, A062503, A286324,

Similar sequences: A005361, A323308, A360721.

Cf. A095691, A365552.

Programs

Mathematica

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

PARI

a(n) = vecprod(apply(x -> if(x%2 || x == 2, x, x-1), factor(n)[, 2]));

Formula

Multiplicative with a(p^e) = e if e = 2 or e is odd, and e-1 otherwise.

a(n) >= 1, with equality if and only if n is squarefree (A005117).

a(n) <= A286324(n), with equality if and only if n equals the square of a squarefree number (A062503).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * Product_{p prime} (1 + 1/p^3 - 1/p^4 + 1/p^5) = 1.87133814920590891161... .