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

1,2

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

Links

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

Example

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.

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 *)

Prog

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

Crossrefs

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

Positions of zeros in A324393.

Sequence in context: A267817 A177917 A071594 * A071596 A090778 A097380

Adjacent sequences: A306260 A306261 A306262 * A306264 A306265 A306266

Keyword

nonn,base

Author

Rémy Sigrist, Mar 02 2019

Status

approved