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

1,4

Comments

First differs from A095691 and A365552 at n = 32.

Links

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

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... .

Mathematica

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

Prog

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

Crossrefs

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

Similar sequences: A005361, A323308, A360721.

Cf. A095691, A365552.

Sequence in context: A360721 A365552 A095691 * A182426 A371733 A067029

Adjacent sequences: A368102 A368103 A368104 * A368106 A368107 A368108

Keyword

nonn,easy,mult

Author

Amiram Eldar, Dec 12 2023

Status

approved