A004754 - OEIS

%I #35 Jul 13 2022 20:37:29

%S 2,4,5,8,9,10,11,16,17,18,19,20,21,22,23,32,33,34,35,36,37,38,39,40,

%T 41,42,43,44,45,46,47,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,

%U 80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,128,129,130,131

%N Numbers n whose binary expansion starts 10.

%C A000120(a(n)) = A000120(n); A023416(a(n-1)) = A008687(n) for n > 1. - _Reinhard Zumkeller_, Dec 04 2015

%H T. D. Noe, <a href="/A004754/b004754.txt">Table of n, a(n) for n = 1..1023</a>

%H Ralf Stephan, <a href="/somedcgf.html">Some divide-and-conquer sequences ...</a>

%H Ralf Stephan, <a href="/A079944/a079944.ps">Table of generating functions</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%F a(2n) = 2a(n), a(2n+1) = 2a(n) + 1 + [n==0].

%F a(n) = n + 2^floor(log_2(n)) = n + A053644(n).

%F a(2^m+k) = 2^(m+1) + k, m >= 0, 0 <= k < 2^m. - _Yosu Yurramendi_, Aug 08 2016

%e 10 in binary is 1010, so 10 is in sequence.

%t w = {1, 0}; Select[Range[2, 131], If[# < 2^(Length@ w - 1), True, Take[IntegerDigits[#, 2], Length@ w] == w] &] (* _Michael De Vlieger_, Aug 08 2016 *)

%o (PARI) a(n)=n+2^floor(log(n)/log(2))

%o (PARI) is(n)=n>1 && !binary(n)[2] \\ _Charles R Greathouse IV_, Sep 23 2012

%o (Haskell)

%o import Data.List (transpose)

%o a004754 n = a004754_list !! (n-1)

%o a004754_list = 2 : concat (transpose [zs, map (+ 1) zs])

%o                    where zs = map (* 2) a004754_list

%o -- _Reinhard Zumkeller_, Dec 04 2015

%o (Python)

%o def A004754(n): return n+(1<<n.bit_length()-1) # _Chai Wah Wu_, Jul 13 2022

%Y Cf. A123001 (binary version), A004755 (11), A004756 (100), A004757 (101), A004758 (110), A004759 (111).

%Y Cf. A004760, A053644, A062050, A076877.

%Y Apart from initial terms, same as A004761.

%Y Cf. A000120, A023416, A008687.

%K nonn,easy,base

%O 1,1

%A _N. J. A. Sloane_

%E Edited by _Ralf Stephan_, Oct 12 2003