A324393 -id:A324393 - cp's OEIS frontend

A306263 Numbers k such that, for any divisor d of k, the Hamming weight of d divides k.

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 24, 32, 34, 36, 40, 42, 48, 60, 64, 66, 68, 72, 80, 84, 92, 96, 108, 116, 120, 126, 128, 132, 136, 144, 156, 160, 168, 172, 180, 184, 192, 204, 212, 216, 222, 228, 232, 240, 246, 252, 256, 264, 272, 276, 284, 288, 300, 310

Offset: 1

Rémy Sigrist, Mar 02 2019

The Hamming weight of a number is given by A000120.
This sequence is a binary variant of A285815.
This sequence is infinite as it contains all powers of 2 (A000079).
All terms belong to A049445.
If k belongs to the sequence, then 2*k belongs to the sequence.
All terms except 1 are even. - Robert Israel, Mar 05 2019

			For n = 108:
- the divisors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108,
- the corresponding Hamming weights are 1, 1, 2, 1, 2, 2, 2, 2, 4, 2, 4, 4,
- they all divide 108,
- hence 108 belongs to the sequence.
For n = 98:
- the divisors of 98 are 1, 2, 7, 14, 49, 98,
- the correspond Hamming weights are 1, 1, 3, 3, 3, 3,
- 3 does not divide 98,
- hence 98 does not belong to the sequence.

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

Cf. A000079, A000120, A049445, A141586, A285815.

Positions of zeros in A324393.

[k:k in [1..310]| forall{d:d in Divisors(k)| k mod &+Intseq(d,2) eq 0}]; // Marius A. Burtea, Dec 30 2019

filter:= proc(n) local F;
  F:= map(convert,map(convert,numtheory:-divisors(n),base,2),`+`);
  andmap(t -> n mod t = 0, F)
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 05 2019

Select[Range@ 310, With[{k = #}, AllTrue[Divisors@ k, Mod[k, DigitCount[#, 2, 1]] == 0 &]] &] (* Michael De Vlieger, Mar 05 2019 *)

is(n) = fordiv(n,d,if (n%hammingweight(d), return (0))); return ( )

A324392 a(n) is the number of divisors d of n such that A000120(d) divides n, where A000120(d) gives the binary weight of d.

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 1, 4, 1, 4, 1, 6, 1, 2, 1, 5, 1, 6, 1, 6, 3, 2, 1, 8, 1, 2, 1, 3, 1, 6, 1, 6, 2, 4, 1, 9, 1, 2, 2, 8, 1, 8, 1, 3, 1, 2, 1, 10, 1, 4, 1, 3, 1, 6, 2, 4, 2, 2, 1, 12, 1, 2, 3, 7, 1, 8, 1, 6, 2, 4, 1, 12, 1, 2, 2, 3, 1, 6, 1, 10, 2, 2, 1, 12, 1, 2, 1, 4, 1, 8, 1, 6, 1, 2, 1, 12, 1, 2, 2, 6, 1, 6, 1, 4, 4

Offset: 1

Antti Karttunen, Mar 05 2019

Number of such positive integers k that both k and A000120(k) [the Hamming weight of k] divide n.

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

Cf. A000005, A000120, A324393.

Cf. also A093653, A192895, A266342, A292257.

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

A324392(n) = sumdiv(n, d, !(n%hammingweight(d)));

a(n) = Sum_{d|n} [A000120(d) does divide n], where [ ] is the Iverson bracket.
a(n) = A000005(n) - A324393(n).
a(p) = 1 for all odd primes p.

cp's OEIS Frontend

A306263 Numbers k such that, for any divisor d of k, the Hamming weight of d divides k.

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