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

%I A180054 #11 Sep 01 2021 12:06:15
%S A180054 11,22,23,27,43,44,46,47,54,55,59,86,87,88,91,92,94,95,107,108,110,
%T A180054 111,118,119,123,171,172,173,174,175,176,179,182,183,184,187,188,190,
%U A180054 191,203,214,215,216,219,220,222,223,235,236,238,239,246,247,251,299,342
%N A180054 In binary expansion, number of 1's in 3n is less than in n.
%C A180054 Or, binary weight of 3n is less than binary weight of n.
%C A180054 Also called the 3-flimsy numbers; see the Stolarsky reference.
%C A180054 If m is here, 2m is too. Hence the "primitive solutions" are all odd (see A134773):
%C 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, 439, 443, 447, 459, 471, 475, 479, 491, 495, ...
%C A180054 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 A180054 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 A180054 G. C. Greubel, <a href="/A180054/b180054.txt">Table of n, a(n) for n = 1..10000</a>
%H A180054 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 A180054 A000120(n) > A000120(3n).
%e A180054 n=11=1011_2, 3n=33=100001_2; or A000120(11)=3, A000120(3*11)=2
%e A180054 n=23=10111_2, 3n=69=1000101_2; or A000120(23)=4, A000120(3*23)=3.
%t A180054 Select[Range[500],Count[IntegerDigits[3#,2],1]<Count[IntegerDigits[ #, 2],1]&]
%t A180054 Select[Range[350],DigitCount[#,2,1]>DigitCount[3#,2,1]&] (* _Harvey P. Dale_, Sep 01 2021 *)
%o A180054 (PARI) for(k=1,350,if(hammingweight(3*k)<hammingweight(k),print1(k,", "))) \\ _Hugo Pfoertner_, Dec 26 2019
%Y A180054 Cf. A000120, A134773, A180055.
%K A180054 base,nonn
%O A180054 1,1
%A A180054 _Zak Seidov_, Aug 08 2010