A180054 -id:A180054 - cp's OEIS frontend

A134773 Primitive elements of A180054.

11, 23, 27, 43, 47, 55, 59, 87, 91, 95, 107, 111, 119, 123, 171, 173, 175, 179, 183, 187, 191, 203, 215, 219, 223, 235, 239, 247, 251, 299, 343, 345, 347, 349, 351, 355, 359, 363, 365, 367, 371, 375, 379, 383, 395, 407, 411, 427, 429, 431, 435

Offset: 1

Zak Seidov, Aug 08 2010

nonn

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.

Original entry on oeis.org

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

Zak Seidov, Aug 08 2010

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, ...

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.

Cf. A000120, A005360, A180054.

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

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

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

A180059 Smallest k such that binary weight of k*n is less than binary weight of k, or zero if no such k exists (for n = powers of 2).

Original entry on oeis.org

0, 0, 11, 0, 13, 11, 23, 0, 29, 13, 47, 11, 79, 23, 47, 0, 61, 29, 27, 13, 55, 47, 23, 11, 41, 79, 19, 23, 53, 47, 95, 0, 125, 61, 59, 29, 111, 27, 55, 13, 25, 55, 143, 47, 47, 23, 47, 11, 209, 41, 91, 79, 29, 19, 149, 23, 575, 53, 139, 47, 277, 95, 191, 0, 253, 125, 107, 61

Offset: 1

Zak Seidov, Aug 08 2010

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000120, A180054, A180055.

s={};Do[Do[If[Count[IntegerDigits[k x,2],1]Zak Seidov, Oct 24 2013 *)

a(n)=if((n<=1)||((n>>valuation(n, 2))==1), 0, my(k=3); while(hammingweight(k*n)>=hammingweight(k), k+=2); k ) \\ Charles R Greathouse IV, Oct 19 2013

A000120(n) > A000120(k*n).

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A134773 Primitive elements of A180054.

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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.

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A180059 Smallest k such that binary weight of k*n is less than binary weight of k, or zero if no such k exists (for n = powers of 2).

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