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A135124 Numbers such that the digital sums in base 2, base 4 and base 8 are all equal.

%I A135124 #27 May 14 2025 19:44:21
%S A135124 1,64,65,4096,4097,4160,4161,262144,262145,262208,262209,266240,
%T A135124 266241,266304,266305,16777216,16777217,16777280,16777281,16781312,
%U A135124 16781313,16781376,16781377,17039360,17039361,17039424,17039425,17043456
%N A135124 Numbers such that the digital sums in base 2, base 4 and base 8 are all equal.
%C A135124 Written as base 64 numbers the sequence is 1,10,11,100,101,110,111,1000,1001, ... (cf. A007088)
%H A135124 Michael De Vlieger, <a href="/A135124/b135124.txt">Table of n, a(n) for n = 1..8192</a>
%H A135124 Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, <a href="https://arxiv.org/abs/2210.10968">Identities and periodic oscillations of divide-and-conquer recurrences splitting at half</a>, arXiv:2210.10968 [cs.DS], 2022, p. 45.
%F A135124 a(n) = (1/2)*Sum_{k=0..floor(log_2(n))} (1-(-1)^floor(n/2^k))*64^k.
%F A135124 G.f.: (1/(1-x))*Sum_{k>=0} 64^k*x^(2^k)/(1+x^(2^k)).
%F A135124 a(n) = A000695(A033045(n)) = A033045(A000695(n)). - _Alan Michael Gómez Calderón_, Mar 27 2025
%e A135124 a(7)=4161, since ds_2(4161 )=ds_4(4161 )=ds_8(4161 ), where ds_x=digital sum base x.
%t A135124 Select[Range[500000], Total[IntegerDigits[#, 2]] == Total[IntegerDigits[#, 4]] == Total[IntegerDigits[#, 8]] &] (* _G. C. Greubel_, Sep 26 2016 *)
%t A135124 With[{k = 64}, Rest@ Map[FromDigits[#, k] &, Tuples[{0, 1}, 5]]] (* _Michael De Vlieger_, Oct 28 2022 *)
%t A135124 Select[Range[171*10^5],Length[Union[Total/@IntegerDigits[#,{2,4,8}]]]==1&] (* _Harvey P. Dale_, May 14 2025 *)
%o A135124 (PARI) a(n) = fromdigits(binary(n),64); \\ _Kevin Ryde_, Apr 02 2025
%Y A135124 Cf. A007953, A054899, A131451, A133620, A133900, A134599, A135100, A135110, A135120, A037308.
%Y A135124 Cf. A000695, A033045.
%K A135124 nonn,base,easy
%O A135124 1,2
%A A135124 _Hieronymus Fischer_, Dec 31 2007, Dec 31 2008
%E A135124 Edited by _N. J. A. Sloane_, Jan 17 2009