A368104 The number of bi-unitary divisors of the powerful part of n.

1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 4, 2, 1, 4, 2, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1

Offset

1,4

Links

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

Formula

a(n) = A286324(A057521(n)).

Multiplicative with a(p^e) = e if e is even or e = 1, 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 is powerful (A001694).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * Product_{p prime} (1 + 2/p^3 - 1/p^4) = 2.12258268547914758409... .

Mathematica

f[p_, e_] := If[e == 1 || EvenQ[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 == 1 || !(x%2), x, x+1), factor(n)[, 2]));

Crossrefs

Cf. A013661, A001694, A005117, A057521, A286324.

Similar sequences: A323308, A357669, A368106.

Sequence in context: A362905 A370771 A368106 * A104404 A351655 A351656

Adjacent sequences: A368101 A368102 A368103 * A368105 A368106 A368107

Keyword

nonn,easy,mult

Author

Amiram Eldar, Dec 12 2023

Status

approved