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

1, 64, 65, 4096, 4097, 4160, 4161, 262144, 262145, 262208, 262209, 266240, 266241, 266304, 266305, 16777216, 16777217, 16777280, 16777281, 16781312, 16781313, 16781376, 16781377, 17039360, 17039361, 17039424, 17039425, 17043456

Offset: 1

Author

Hieronymus Fischer, Dec 31 2007, Dec 31 2008

Keywords

• nonn
• base
• easy

Comments

Written as base 64 numbers the sequence is 1,10,11,100,101,110,111,1000,1001, ... (cf. A007088)

Examples

			a(7)=4161, since ds_2(4161 )=ds_4(4161 )=ds_8(4161 ), where ds_x=digital sum base x.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..8192
• Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 45.

Crossrefs

Cf. A007953, A054899, A131451, A133620, A133900, A134599, A135100, A135110, A135120, A037308.

Cf. A000695, A033045.

Programs

Mathematica

Select[Range[500000], Total[IntegerDigits[#, 2]] == Total[IntegerDigits[#, 4]] == Total[IntegerDigits[#, 8]] &] (* G. C. Greubel, Sep 26 2016 *)
With[{k = 64}, Rest@ Map[FromDigits[#, k] &, Tuples[{0, 1}, 5]]] (* Michael De Vlieger, Oct 28 2022 *)
Select[Range[171*10^5],Length[Union[Total/@IntegerDigits[#,{2,4,8}]]]==1&] (* Harvey P. Dale, May 14 2025 *)

PARI

a(n) = fromdigits(binary(n),64); \\ Kevin Ryde, Apr 02 2025

Formula

a(n) = (1/2)*Sum_{k=0..floor(log_2(n))} (1-(-1)^floor(n/2^k))*64^k.

G.f.: (1/(1-x))*Sum_{k>=0} 64^k*x^(2^k)/(1+x^(2^k)).

a(n) = A000695(A033045(n)) = A033045(A000695(n)). - Alan Michael Gómez Calderón, Mar 27 2025

Extensions

Edited by N. J. A. Sloane, Jan 17 2009