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A180054 In binary expansion, number of 1's in 3n is less than in n. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 26 06:08 EDT 2021. Contains 348257 sequences. (Running on oeis4.)