cp's OEIS Frontend

A180055 Numbers k such that in binary expansion, the number of 1's in 5*k is less than the number of 1's in k.

%I A180055 #19 Jul 30 2025 09:47:04
%S A180055 13,26,29,52,53,55,58,61,77,103,104,106,109,110,111,116,117,119,122,
%T A180055 125,154,157,205,206,207,208,212,213,215,218,219,220,221,222,223,231,
%U A180055 232,234,237,238,239,244,245,247,250,253,308,309,311,314,317,333,359,365
%N A180055 Numbers k such that in binary expansion, the number of 1's in 5*k is less than the number of 1's in k.
%C A180055 Or, binary weight of 5*k is less than binary weight of k.
%C A180055 Numbers k such that A000120(k) > A000120(5*k).
%C A180055 Also called the 5-flimsy numbers; see the Stolarsky reference.
%C A180055 If m is here, 2m is too. Hence the "primitive solutions" are all odd ones: 13,29,53,55,61,77,103,109,111,117,119,125,157,205,207,213,215,219,221,223,231, ...
%H A180055 Amiram Eldar, <a href="/A180055/b180055.txt">Table of n, a(n) for n = 1..10000</a>
%H A180055 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.
%p A180055 filter:= proc(k) convert(convert(5*k,base,2),`+`) < convert(convert(k,base,2),`+`) end proc:
%p A180055 select(filter, [$1..1000]); # _Robert Israel_, Jul 29 2025
%t A180055 Select[Range[1000],Count[IntegerDigits[5#,2],1]<Count[IntegerDigits[ #,2],1]&]
%t A180055 Select[Range[365], Less@@ DigitSum[{5*#, #}, 2]&] (* _Amiram Eldar_, Jul 29 2025 *)
%o A180055 (PARI) for(k=1,370, if(hammingweight(5*k) < hammingweight(k), print1(k,", "))) \\ _Hugo Pfoertner_, Dec 27 2019
%Y A180055 Cf. A000120, A005360, A180054.
%K A180055 base,nonn,easy
%O A180055 1,1
%A A180055 _Zak Seidov_, Aug 08 2010