A180055 -id:A180055 - cp's OEIS frontend

A180054 In binary expansion, number of 1's in 3n is less than in n.

11, 22, 23, 27, 43, 44, 46, 47, 54, 55, 59, 86, 87, 88, 91, 92, 94, 95, 107, 108, 110, 111, 118, 119, 123, 171, 172, 173, 174, 175, 176, 179, 182, 183, 184, 187, 188, 190, 191, 203, 214, 215, 216, 219, 220, 222, 223, 235, 236, 238, 239, 246, 247, 251, 299, 342

Offset: 1

Zak Seidov, Aug 08 2010

Or, binary weight of 3n is less than binary weight of n.
Also called the 3-flimsy numbers; see the Stolarsky reference.
If m is here, 2m is too. Hence the "primitive solutions" are all odd (see A134773):
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, 439, 443, 447, 459, 471, 475, 479, 491, 495, ...
These are also the cases where A000120(n) > A000120(6*n) because 6*n = 2*(3*n) means that the number of 1's in 6*n and 3*n are the same. - R. J. Mathar, Aug 13 2010
These are also the cases where A000120(n*2^k1) > A000120(3n*2^k2) for any integers k1, k2 >= 0. - Zak Seidov, Aug 15 2010

			n=11=1011_2, 3n=33=100001_2; or A000120(11)=3, A000120(3*11)=2
n=23=10111_2, 3n=69=1000101_2; or A000120(23)=4, A000120(3*23)=3.

G. C. Greubel, 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, A134773, A180055.

Select[Range[500],Count[IntegerDigits[3#,2],1]DigitCount[3#,2,1]&] (* Harvey P. Dale, Sep 01 2021 *)

for(k=1,350,if(hammingweight(3*k)Hugo Pfoertner, Dec 26 2019

A000120(n) > A000120(3n).

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).

cp's OEIS Frontend

A180054 In binary expansion, number of 1's in 3n is less than in n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula