A368106 The number of infinitary divisors of the powerful part of n.

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

Offset

1,4

Links

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

Formula

a(n) = A037445(A057521(n)).

Multiplicative with a(p) = 1 and a(p^e) = 2^A000120(e) for e >= 2.

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

a(n) <= A037445(n), with equality if and only if n is powerful (A001694).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.89684906463124350536..., where f(x) = (1-x) * (Product_{k>=0} (1 + 2*x^(2^k)) - x).

Mathematica

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

Prog

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

Crossrefs

Cf. A001694, A005117, A037445, A057521.

Similar sequences: A323308, A357669, A368104.

Sequence in context: A173398 A362905 A370771 * A368104 A104404 A351655

Adjacent sequences: A368103 A368104 A368105 * A368107 A368108 A368109

Keyword

nonn,easy,mult

Author

Amiram Eldar, Dec 12 2023

Status

approved