A180054 - OEIS

%I #11 Sep 01 2021 12:06:15

%S 11,22,23,27,43,44,46,47,54,55,59,86,87,88,91,92,94,95,107,108,110,

%T 111,118,119,123,171,172,173,174,175,176,179,182,183,184,187,188,190,

%U 191,203,214,215,216,219,220,222,223,235,236,238,239,246,247,251,299,342

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

%C Or, binary weight of 3n is less than binary weight of n.

%C Also called the 3-flimsy numbers; see the Stolarsky reference.

%C If m is here, 2m is too. Hence the "primitive solutions" are all odd (see A134773):

%C 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, ...

%C 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

%C 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

%H G. C. Greubel, <a href="/A180054/b180054.txt">Table of n, a(n) for n = 1..10000</a>

%H Kenneth B. Stolarsky, <a href="https://eudml.org/doc/205727">Integers whose multiples have anomalous digital frequencies</a>, Acta Arithmetica 38 (2) (1980), 117-128.

%F A000120(n) > A000120(3n).

%e n=11=1011_2, 3n=33=100001_2; or A000120(11)=3, A000120(3*11)=2

%e n=23=10111_2, 3n=69=1000101_2; or A000120(23)=4, A000120(3*23)=3.

%t Select[Range[500],Count[IntegerDigits[3#,2],1]<Count[IntegerDigits[ #, 2],1]&]

%t Select[Range[350],DigitCount[#,2,1]>DigitCount[3#,2,1]&] (* _Harvey P. Dale_, Sep 01 2021 *)

%o (PARI) for(k=1,350,if(hammingweight(3*k)<hammingweight(k),print1(k,", "))) \\ _Hugo Pfoertner_, Dec 26 2019

%Y Cf. A000120, A134773, A180055.

%K base,nonn

%O 1,1

%A _Zak Seidov_, Aug 08 2010