A372504 Multiplicative with a(p^e) = e if e is a power of 2, and 0 otherwise.

1, 1, 1, 2, 1, 1, 1, 0, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 0, 2, 1, 0, 2, 1, 1, 1, 0, 1, 1, 1, 4, 1, 1, 1, 0, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 0, 1, 0, 1, 1, 1, 2, 1, 1, 2, 0, 1, 1, 1, 2, 1, 1, 1, 0, 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

Multiplicative with a(p^e) = A048298(e) = A209229(e) * e.

a(n) = A355823(n) * A005361(n).

a(A138302(n)) = A005361(A138302(n)) = A368473(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.31285540951965780409..., where f(x) = (1-x) * (1 + Sum_{k>=0} 2^k * x^(2^k)).

Mathematica

f[p_, e_] := If[e == 2^IntegerExponent[e, 2], e, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = vecprod(apply(x -> if(x == 1 << valuation(x, 2), x, 0), factor(n)[, 2]));

Crossrefs

Cf. A005361, A048298, A138302, A209229, A355823, A368473, A372470.

Sequence in context: A146061 A331186 A375847 * A135936 A109707 A214578

Adjacent sequences: A372501 A372502 A372503 * A372505 A372506 A372507

Keyword

nonn,easy,mult

Author

Amiram Eldar, May 04 2024

Status

approved