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A368104
The number of bi-unitary divisors of the powerful part of n.
2
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
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
Similar sequences: A323308, A357669, A368106.
Sequence in context: A362905 A370771 A368106 * A104404 A351655 A351656
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Dec 12 2023
STATUS
approved