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

1,1

Comments

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

Links

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.

Formula

A000120(n) > A000120(3n).

Example

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.

Mathematica

Select[Range[500], Count[IntegerDigits[3#, 2], 1]<Count[IntegerDigits[ #, 2], 1]&]
Select[Range[350], DigitCount[#, 2, 1]>DigitCount[3#, 2, 1]&] (* Harvey P. Dale, Sep 01 2021 *)

Prog

(PARI) for(k=1, 350, if(hammingweight(3*k)<hammingweight(k), print1(k, ", "))) \\ Hugo Pfoertner, Dec 26 2019

Crossrefs

Cf. A000120, A134773, A180055.

Sequence in context: A040110 A140452 A048136 * A166713 A297273 A296746

Adjacent sequences: A180051 A180052 A180053 * A180055 A180056 A180057

Keyword

base,nonn

Author

Zak Seidov, Aug 08 2010

Status

approved