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

Author

Rémy Sigrist, Mar 02 2019

Keywords

• nonn
• base

Comments

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

Examples

			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.
		

Links

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

Crossrefs

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

Positions of zeros in A324393.

Programs

Magma

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

Maple

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

Mathematica

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

PARI

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