A324393 a(n) is the number of such divisors d of n that A000120(d) does not divide n, where A000120(d) gives the binary weight of d.

0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 2, 3, 0, 1, 0, 1, 0, 1, 2, 1, 0, 2, 2, 3, 3, 1, 2, 1, 0, 2, 0, 3, 0, 1, 2, 2, 0, 1, 0, 1, 3, 5, 2, 1, 0, 2, 2, 3, 3, 1, 2, 2, 4, 2, 2, 1, 0, 1, 2, 3, 0, 3, 0, 1, 0, 2, 4, 1, 0, 1, 2, 4, 3, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 3, 4, 1, 4, 3, 0, 3, 2, 3, 0, 1, 4, 4, 3, 1, 2, 1, 4, 4

Offset: 1

Author

Antti Karttunen, Mar 05 2019

Keywords

• nonn
• base

Comments

Number of such positive integers k that divide n but A000120(k) [the Hamming weight of k] does not divide n.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A000005, A000120, A324392, A306263 (positions of zeros).

Cf. also A093653, A192895, A266342, A292257.

Programs

Mathematica

a[n_] := DivisorSum[n, 1 &, !Divisible[n, DigitCount[#, 2, 1]] &]; Array[a, 100] (* Amiram Eldar, Dec 04 2020 *)

PARI

A324393(n) = sumdiv(n, d, !!(n%hammingweight(d)));

Formula

a(n) = Sum_{d|n} [A000120(d) does not divide n], where [ ] is the Iverson bracket.

a(n) = A000005(n) - A324392(n).

a(p) = 1 for all odd primes p.