A180055 Numbers k such that in binary expansion, the number of 1's in 5*k is less than the number of 1's in k.

13, 26, 29, 52, 53, 55, 58, 61, 77, 103, 104, 106, 109, 110, 111, 116, 117, 119, 122, 125, 154, 157, 205, 206, 207, 208, 212, 213, 215, 218, 219, 220, 221, 222, 223, 231, 232, 234, 237, 238, 239, 244, 245, 247, 250, 253, 308, 309, 311, 314, 317, 333, 359, 365

Offset: 1

Author

Zak Seidov, Aug 08 2010

Keywords

• base
• nonn
• easy

Comments

Or, binary weight of 5*k is less than binary weight of k.

Numbers k such that A000120(k) > A000120(5*k).

Also called the 5-flimsy numbers; see the Stolarsky reference.

If m is here, 2m is too. Hence the "primitive solutions" are all odd ones: 13,29,53,55,61,77,103,109,111,117,119,125,157,205,207,213,215,219,221,223,231, ...

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Kenneth B. Stolarsky, Integers whose multiples have anomalous digital frequencies, Acta Arithmetica 38 (2) (1980), 117-128.

Crossrefs

Cf. A000120, A005360, A180054.

Programs

Maple

filter:= proc(k) convert(convert(5*k,base,2),`+`) < convert(convert(k,base,2),`+`) end proc:
select(filter, [$1..1000]); # Robert Israel, Jul 29 2025

Mathematica

Select[Range[1000],Count[IntegerDigits[5#,2],1]Amiram Eldar, Jul 29 2025 *)

PARI

for(k=1,370, if(hammingweight(5*k) < hammingweight(k), print1(k,", "))) \\ Hugo Pfoertner, Dec 27 2019