A317946 Additive with a(p^e) = A011371(e); the 2-adic valuation of A317934(n).

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 3, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 4, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 3, 3, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 3, 0, 1, 1, 2, 0, 0, 0, 1, 0

Offset: 1

Author

Antti Karttunen, Aug 24 2018

Keywords

• nonn

Comments

Records are A005187, occurring at A000302 (powers of 4).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A000302, A005187, A011371, A317934.

Cf. also A046645.

Programs

Mathematica

f[p_, e_] := e - DigitCount[e, 2, 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 21 2024 *)

PARI

A011371(n) = (n - hammingweight(n));
A317934(n) = vecsum(apply(e -> A011371(e),factor(n)[,2]));

Formula

a(n) = A007814(A317934(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime, k>=1} 1/(p^(2^k) - 1) = 0.63710219855356676263... . - Amiram Eldar, Jan 21 2024