A000523 -id:A000523 - cp's OEIS frontend

A036987 Fredholm-Rueppel sequence.

1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

Binary representation of the Kempner-Mahler number Sum_{k>=0} 1/2^(2^k) = A007404.
Also a(n) = A(n) mod 2 where A is any of A001700, A005573, A007854, A026641, A049027, A064063, A064088, A064090, A064092, A064325, A064327, A064329, A064331, A064613, A076026, A105523, A123273, A126694, A126930, A126931, A126982, A126983, A126987, A127016, A127053, A127358, A127360, A127361, A127363. - Philippe Deléham, May 26 2007
a(n) = (product of digits of n; n in binary notation) mod 2. This sequence is a transformation of the Thue-Morse sequence (A010060), since there exists a function f such that f(sum of digits of n) = (product of digits of n). - Ctibor O. Zizka, Feb 12 2008
a(n-1), n >= 1, the characteristic sequence for powers of 2, A000079, is the unique solution of the following formal product and formal power series identity: Product_{j>=1} (1 + a(j-1)*x^j) = 1 + Sum_{k>=1} x^k = 1/(1-x). The product is therefore Product_{l>=1} (1 + x^(2^l)). Proof. Compare coefficients of x^n and use the binary representation of n. Uniqueness follows from the recurrence relation given for the general case under A147542. - Wolfdieter Lang, Mar 05 2009
a(n) is also the number of orbits of length n for the map x -> 1-cx^2 on [-1,1] at the Feigenbaum critical value c=1.401155... . - Thomas Ward, Apr 08 2009
A054525 (Mobius transform) * A001511 = A036987 = A047999^(-1) * A001511 = the inverse of Sierpiński's gasket * the ruler sequence. - Gary W. Adamson, Oct 26 2009 [Of course this is only vaguely correct depending on how the fuzzy indexing in these formulas is made concrete. - R. J. Mathar, Jun 20 2014]
Characteristic function of A000225. - Reinhard Zumkeller, Mar 06 2012
Also parity of the Catalan numbers A000108. - Omar E. Pol, Jan 17 2012
For n >= 2, also the largest exponent k >= 0 such that n^k in binary notation does not contain both 0 and 1. Unlike for the decimal version of this sequence, A062518, where the terms are only conjectural, for this sequence the values of a(n) can be proved to be the characteristic function of A000225, as follows: n^k will contain both 0 and 1 unless n^k = 2^r-1 for some r. But this is a special case of Catalan's equation x^p = y^q-1, which was proved by Preda Mihăilescu to have no nontrivial solution except 2^3 = 3^2 - 1. - Christopher J. Smyth, Aug 22 2014
Image, under the coding a,b -> 1; c -> 0, of the fixed point, starting with a, of the morphism a -> ab, b -> cb, c -> cc. - Jeffrey Shallit, May 14 2016
Number of nonisomorphic Boolean algebras of order n+1. - Jianing Song, Jan 23 2020

			G.f. = 1 + x + x^3 + x^7 + x^15 + x^31 + x^63 + x^127 + x^255 + x^511 + ...
a(7) = 1 since 7 = 2^3 - 1, while a(10) = 0 since 10 is not of the form 2^k - 1 for any integer k.

Antti Karttunen, Table of n, a(n) for n = 0..65537
D. Bailey et al., On the binary expansions of algebraic numbers, Journal de Théorie des Nombres de Bordeaux 16 (2004), 487-518.
Paul Barry, Some observations on the Rueppel sequence and associated Hankel determinants, arXiv:2005.04066 [math.CO], 2020.
Paul Barry, Conjectures and results on some generalized Rueppel sequences, arXiv:2107.00442 [math.CO], 2021.
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.
Mats Granvik, Mathematica program for computing Fredholm Rueppel sequences
D. Kohel, S. Ling and C. Xing, Explicit Sequence Expansions, in Sequences and their Applications, C. Ding, T. Helleseth, and H. Niederreiter, eds., Proceedings of SETA'98 (Singapore, 1998), 308-317, 1999.
Preda Mihăilescu, Primary Cyclotomic Units and a Proof of Catalan's Conjecture, J. Reine angew. Math. 572 (2004): 167-195. doi:10.1515/crll.2004.048. MR 2076124.
H. Niederreiter and M. Vielhaber, Tree Complexity and a Doubly Exponential Gap between Structured and Random Sequences, J. Complexity, 12 (1996), 187-198.
Apisit Pakapongpun and Thomas Ward, Functorial orbit counting, Journal of Integer Sequences, 12 (2009) Article 09.2.4. [From _Thomas Ward_, Apr 08 2009]
Eric Rowland and Reem Yassawi, Profinite automata, arXiv:1403.7659 [math.DS], 2014. See p. 8.
E. Sheppard, net.math post (1985)
Stephen Wolfram, [Page 1092] A New Kind of Science | Online.
Index entries for characteristic functions

Cf. A007404, A043545, A062518, A078885, A078585, A078886, A078887, A078888, A078889, A078890.
The first row of A073346. Occurs for first time in A073202 as row 6 (and again as row 8).
Congruent to any of the sequences A000108, A007460, A007461, A007463, A007464, A061922, A068068 reduced modulo 2. Characteristic function of A000225.
If interpreted with offset=1 instead of 0 (i.e., a(1)=1, a(2)=1, a(3)=0, a(4)=1, ...) then this is the characteristic function of 2^n (A000079) and as such occurs as the first row of A073265. Also, in that case the INVERT transform will produce A023359.
This is Guy Steele's sequence GS(1, 3), also GS(3, 1) (see A135416).
Cf. A054525, A047999. - Gary W. Adamson, Oct 26 2009
Cf. A043529, A127802.

a036987 n = ibp (n+1) where
   ibp 1 = 1
   ibp n = if r > 0 then 0 else ibp n' where (n',r) = divMod n 2
a036987_list = 1 : f [0,1] where f (x:y:xs) = y : f (x:xs ++ [x,x+y])
-- Same list generator function as for a091090_list, cf. A091090.
-- Reinhard Zumkeller, May 19 2015, Apr 13 2013, Mar 13 2013

A036987:= n-> `if`(2^ilog2(n+1) = n+1, 1, 0):
seq(A036987(n), n=0..128);

RealDigits[ N[ Sum[1/10^(2^n), {n, 0, Infinity}], 110]][[1]]
(* Recurrence: *)
t[n_, 1] = 1; t[1, k_] = 1;
t[n_, k_] := t[n, k] =
  If[n < k, If[n > 1 && k > 1, -Sum[t[k - i, n], {i, 1, n - 1}], 0],
   If[n > 1 && k > 1, Sum[t[n - i, k], {i, 1, k - 1}], 0]];
Table[t[n, k], {k, n, n}, {n, 104}]
(* Mats Granvik, Jun 03 2011 *)
mb2d[n_]:=1 - Module[{n2 = IntegerDigits[n, 2]}, Max[n2] - Min[n2]]; Array[mb2d, 120, 0] (* Vincenzo Librandi, Jul 19 2019 *)
Table[PadRight[{1},2^k,0],{k,0,7}]//Flatten (* Harvey P. Dale, Apr 23 2022 *)

{a(n) =( n++) == 2^valuation(n, 2)}; /* Michael Somos, Aug 25 2003 */

a(n) = !bitand(n, n+1); \\ Ruud H.G. van Tol, Apr 05 2023

from sympy import catalan
def a(n): return catalan(n)%2 # Indranil Ghosh, May 25 2017

def A036987(n): return int(not(n&(n+1))) # Chai Wah Wu, Jul 06 2022

1 followed by a string of 2^k - 1 0's. Also a(n)=1 iff n = 2^m - 1.
a(n) = a(floor(n/2)) * (n mod 2) for n>0 with a(0)=1. - Reinhard Zumkeller, Aug 02 2002 [Corrected by Mikhail Kurkov, Jul 16 2019]
Sum_{n>=0} 1/10^(2^n) = 0.110100010000000100000000000000010...
1 if n=0, floor(log_2(n+1)) - floor(log_2(n)) otherwise. G.f.: (1/x) * Sum_{k>=0} x^(2^k) = Sum_{k>=0} x^(2^k-1). - Ralf Stephan, Apr 28 2003
a(n) = 1 - A043545(n). - Michael Somos, Aug 25 2003
a(n) = -Sum_{d|n+1} mu(2*d). - Benoit Cloitre, Oct 24 2003
Dirichlet g.f. for right-shifted sequence: 2^(-s)/(1-2^(-s)).
a(n) = A000108(n) mod 2 = A001405(n) mod 2. - Paul Barry, Nov 22 2004
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*Sum_{j=0..k} binomial(k, 2^j-1). - Paul Barry, Jun 01 2006
A000523(n+1) = Sum_{k=1..n} a(k). - Mitch Harris, Jul 22 2011
a(n) = A209229(n+1). - Reinhard Zumkeller, Mar 07 2012
a(n) = Sum_{k=1..n} A191898(n,k)*cos(Pi*(n-1)*(k-1))/n; (conjecture). - Mats Granvik, Mar 04 2013
a(n) = A000035(A000108(n)). - Omar E. Pol, Aug 06 2013
a(n) = 1 iff n=2^k-1 for some k, 0 otherwise. - M. F. Hasler, Jun 20 2014
a(n) = ceiling(log_2(n+2)) - ceiling(log_2(n+1)). - Gionata Neri, Sep 06 2015
From John M. Campbell, Jul 21 2016: (Start)
a(n) = (A000168(n-1) mod 2).
a(n) = (A000531(n+1) mod 2).
a(n) = (A000699(n+1) mod 2).
a(n) = (A000891(n) mod 2).
a(n) = (A000913(n-1) mod 2), for n>1.
a(n) = (A000917(n-1) mod 2), for n>0.
a(n) = (A001142(n) mod 2).
a(n) = (A001246(n) mod 2).
a(n) = (A001246(n) mod 4).
a(n) = (A002057(n-2) mod 2), for n>1.
a(n) = (A002430(n+1) mod 2). (End)
a(n) = 2 - A043529(n). - Antti Karttunen, Nov 19 2017
a(n) = floor(1+log(n+1)/log(2)) - floor(log(2n+1)/log(2)). - Adriano Caroli, Sep 22 2019
This is also the decimal expansion of -Sum_{k>=1} mu(2*k)/(10^k - 1), where mu is the Möbius function (A008683). - Amiram Eldar, Jul 12 2020

Edited by M. F. Hasler, Jun 20 2014

A053644 Most significant bit of n, msb(n); largest power of 2 less than or equal to n; write n in binary and change all but the first digit to zero.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64

Offset: 0

Henry Bottomley, Mar 22 2000

Except for the initial term, 2^n appears 2^n times. - Lekraj Beedassy, May 26 2005
a(n) is the smallest k such that row k in triangle A265705 contains n. - Reinhard Zumkeller, Dec 17 2015
a(n) is the sum of totient function over powers of 2 <= n. - Anthony Browne, Jun 17 2016
Given positive n, reverse the bits of n and divide by 2^floor(log_2 n). Numerators are in A030101. Ignoring the initial 0, denominators are in this sequence. - Alonso del Arte, Feb 11 2020

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions

See A000035 for least significant bit(n).
MASKTRANS transform of A055975 (prepended with 0), MASKTRANSi transform of A048678.
Bisection of A065267, A065279, A065291, A072376.
First differences of A063915. Cf. A076877, A073121.
This is Guy Steele's sequence GS(5, 5) (see A135416).
Equals for n >= 1 the first right hand column of A160464. - Johannes W. Meijer, May 24 2009
Diagonal of A088370. - Alois P. Heinz, Oct 28 2011
Cf. A265705, A000010.

a053644 n = if n <= 1 then n else 2 * a053644 (div n 2)
-- Reinhard Zumkeller, Aug 28 2014
a053644_list = 0 : concat (iterate (\zs -> map (* 2) (zs ++ zs)) [1])
-- Reinhard Zumkeller, Dec 08 2012, Oct 21 2011, Oct 17 2010

[0] cat [2^Ilog2(n): n in [1..90]]; // Vincenzo Librandi, Dec 11 2018

a:= n-> 2^ilog2(n):
seq(a(n), n=0..80);  # Alois P. Heinz, Dec 20 2016

A053644[n_] := 2^(Length[ IntegerDigits[n, 2]] - 1); A053644[0] = 0; Table[A053644[n], {n, 0, 74}] (* Jean-François Alcover, Dec 01 2011 *)
nv[n_] := Module[{c = 2^n}, Table[c, {c}]]; Join[{0}, Flatten[Array[nv, 7, 0]]] (* Harvey P. Dale, Jul 17 2012 *)

a(n)=my(k=1);while(k<=n,k<<=1);k>>1 \\ Charles R Greathouse IV, May 27 2011

a(n) = if(!n, 0, 2^exponent(n)) \\ Iain Fox, Dec 10 2018

def a(n): return 0 if n==0 else 2**(len(bin(n)[2:]) - 1) # Indranil Ghosh, May 25 2017

def A053644(n): return 1<Chai Wah Wu, Jul 27 2022

(0 to 127).map(Integer.highestOneBit()) // _Alonso del Arte, Feb 26 2020

a(n) = a(floor(n / 2)) * 2.
a(n) = 2^A000523(n).
From n >= 1 onward, A053644(n) = A062383(n)/2.
a(0) = 0, a(1) = 1 and a(n+1) = a(n)*floor(n/a(n)). - Benoit Cloitre, Aug 17 2002
G.f.: 1/(1 - x) * (x + Sum_{k >= 1} 2^(k - 1)*x^2^k). - Ralf Stephan, Apr 18 2003
a(n) = (A003817(n) + 1)/2 = A091940(n) + 1. - Reinhard Zumkeller, Feb 15 2004
a(n) = Sum_{k = 1..n} (floor(2^k/k) - floor((2^k - 1)/k))*A000010(k). - Anthony Browne, Jun 17 2016
a(2^m+k) = 2^m, m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Aug 07 2016

A059893 Reverse the order of all but the most significant bit in binary expansion of n: if n = 1ab..yz then a(n) = 1zy..ba.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 7, 8, 12, 10, 14, 9, 13, 11, 15, 16, 24, 20, 28, 18, 26, 22, 30, 17, 25, 21, 29, 19, 27, 23, 31, 32, 48, 40, 56, 36, 52, 44, 60, 34, 50, 42, 58, 38, 54, 46, 62, 33, 49, 41, 57, 37, 53, 45, 61, 35, 51, 43, 59, 39, 55, 47, 63, 64, 96, 80, 112, 72, 104, 88, 120

Offset: 1

Marc LeBrun, Feb 06 2001

A self-inverse permutation of the natural numbers.
a(n)=n if and only if A081242(n) is a palindrome. - Clark Kimberling, Mar 12 2003
a(n) is the position in B of the reversal of the n-th term of B, where B is the left-to-right binary enumeration sequence (A081242 with the empty word attached as first term). - Clark Kimberling, Mar 12 2003
From Antti Karttunen, Oct 28 2001: (Start)
When certain Stern-Brocot tree-related permutations are conjugated with this permutation, they induce a permutation on Z (folded to N), which is an infinite siteswap permutation (see, e.g., figure 7 in the Buhler and Graham paper, which is permutation A065174). We get:
   A065260(n) = a(A057115(a(n))),
   A065266(n) = a(A065264(a(n))),
   A065272(n) = a(A065270(a(n))),
   A065278(n) = a(A065276(a(n))),
   A065284(n) = a(A065282(a(n))),
   A065290(n) = a(A065288(a(n))). (End)
Every nonnegative integer has a unique representation c(1) + c(2)*2 + c(3)*2^2 + c(4)*2^3 + ..., where every c(i) is 0 or 1. Taking tuples of coefficients in lexical order (i.e., 0, 1; 01,11; 001,011,101,111; ...) yields A059893. - Clark Kimberling, Mar 15 2015
From Ed Pegg Jr, Sep 09 2015: (Start)
The reduced rationals can be ordered either as the Calkin-Wilf tree A002487(n)/A002487(n+1) or the Stern-Brocot tree A007305(n+2)/A047679(n). The present sequence gives the order of matching rationals in the other sequence.
For reference, the Calkin-Wilf tree is 1, 1/2, 2, 1/3, 3/2, 2/3, 3, 1/4, 4/3, 3/5, 5/2, 2/5, 5/3, 3/4, 4, 1/5, 5/4, 4/7, 7/3, 3/8, 8/5, 5/7, 7/2, 2/7, 7/5, 5/8, 8/3, 3/7, 7/4, 4/5, ..., which is A002487(n)/A002487(n+1).
The Stern-Brocot tree is 1, 1/2, 2, 1/3, 2/3, 3/2, 3, 1/4, 2/5, 3/5, 3/4, 4/3, 5/3, 5/2, 4, 1/5, 2/7, 3/8, 3/7, 4/7, 5/8, 5/7, 4/5, 5/4, 7/5, 8/5, 7/4, 7/3, 8/3, 7/2, ..., which is A007305(n+2)/A047679(n).
There is a great little OEIS-is-useful story here. I had code for the position of fractions in the Calkin-Wilf tree. The best I had for positions of fractions in the Stern-Brocot tree was the paper "Locating terms in the Stern-Brocot tree" by Bruce Bates, Martin Bunder, Keith Tognetti. The method was opaque to me, so I used my Calkin-Wilf code on the Stern-Brocot fractions, and got A059893. And thus the problem was solved. (End)

			a(11) = a(1011) = 1110 = 14.
With empty word e prefixed, A081242 becomes (e,1,2,11,21,12,22,111,211,121,221,112,...); (reversal of term #9) = (term #12); i.e., a(9)=12 and a(12)=9. - _Clark Kimberling_, Mar 12 2003
From _Philippe Deléham_, Jun 02 2015: (Start)
This sequence regarded as a triangle with rows of lengths 1, 2, 4, 8, 16, ...:
   1;
   2,  3;
   4,  6,  5,  7;
   8, 12, 10, 14,  9,  13,  11,  15;
  16, 24, 20, 28, 18,  26,  22,  30,  17,  25,  21,  29,  19,  27,  23,  31;
  32, 48, 40, 56, 36,  52,  44, ...
Row sums = A010036. (End)

Alois P. Heinz, Table of n, a(n) for n = 1..8191 (first 1023 terms from T. D. Noe)
Bruce Bates, Martin Bunder, and Keith Tognetti, Locating terms in the Stern-Brocot tree, European Journal of Combinatorics 31.3 (2010): 1020-1033.
Joe Buhler and R. L. Graham, Juggling Drops and Descents, Amer. Math. Monthly, 101, (no. 6) 1994, 507-519.
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, 2020-2021.
Wikipedia, Calkin-Wilf Tree.
Wikipedia, Stern-Brocot tree.
Index entries for sequences related to Stern's sequences
Index entries for sequences that are permutations of the natural numbers

{A000027, A054429, A059893, A059894} form a 4-group.
The set of permutations {A059893, A080541, A080542} generates an infinite dihedral group.
Cf. A000079, A000523, A007931, A030308.
In other bases: A351702 (balanced ternary), A343150 (Zeckendorf), A343152 (lazy Fibonacci).

a059893 = foldl (\v b -> v * 2 + b) 1 . init . a030308_row
-- Reinhard Zumkeller, May 01 2013
(Scheme, with memoization-macro definec)
(definec (A059893 n) (if (<= n 1) n (let* ((k (- (A000523 n) 1)) (r (A059893 (- n (A000079 k))))) (if (= 2 (floor->exact (/ n (A000079 k)))) (* 2 r) (+ 1 r)))))
;; Antti Karttunen, May 16 2015

# Implements Bottomley's formula
A059893 := proc(n) option remember; local k; if(1 = n) then RETURN(1); fi; k := floor_log_2(n)-1; if(2 = floor(n/(2^k))) then RETURN(2*A059893(n-(2^k))); else RETURN(1+A059893(n-(2^k))); fi; end;
floor_log_2 := proc(n) local nn,i; nn := n; for i from -1 to n do if(0 = nn) then RETURN(i); fi; nn := floor(nn/2); od; end;
# second Maple program:
a:= proc(n) local i, m, r; m, r:= n, 0;
      for i from 0 while m>1 do r:= 2*r +irem(m,2,'m') od;
      r +2^i
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 28 2015

A059893 = Reap[ For[n=1, n <= 100, n++, a=1; b=n; While[b > 1, a = 2*a + 2*FractionalPart[b/2]; b=Floor[b/2]]; Sow[a]]][[2, 1]] (* Jean-François Alcover, Jul 16 2012, after Harry J. Smith *)
ro[n_]:=Module[{idn=IntegerDigits[n,2]},FromDigits[Join[{First[idn]}, Reverse[ Rest[idn]]],2]]; Array[ro,80] (* Harvey P. Dale, Oct 24 2012 *)

a(n) = my(b=binary(n)); fromdigits(concat(b[1], Vecrev(vector(#b-1, k, b[k+1]))), 2); \\ Michel Marcus, Sep 29 2021

def a(n): return int('1' + bin(n)[3:][::-1], 2)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Mar 21 2017

maxrow <- 6 # by choice
a <- 1
for(m in 0:maxrow) for(k in 0:(2^m-1)) {
  a[2^(m+1)+    k] <- 2*a[2^m+k]
  a[2^(m+1)+2^m+k] <- 2*a[2^m+k] + 1
}
a
# Yosu Yurramendi, Mar 20 2017

maxblock <- 7 # by choice
a <- 1
for(n in 2:2^maxblock){
  ones <- which(as.integer(intToBits(n)) == 1)
  nbit <- as.integer(intToBits(n))[1:tail(ones, n = 1)]
  anbit <- nbit
  anbit[1:(length(anbit) - 1)] <- anbit[rev(1:(length(anbit)-1))]
  a <- c(a, sum(anbit*2^(0:(length(anbit) - 1))))
}
a
# Yosu Yurramendi, Apr 25 2021

a(n) = A030109(n) + A053644(n). If 2*2^k <= n < 3*2^k then a(n) = 2*a(n-2^k); if 3*2^k <= n < 4*2^k then a(n) = 1 + a(n-2^k) starting with a(1)=1. - Henry Bottomley, Sep 13 2001

A053645 Distance to largest power of 2 less than or equal to n; write n in binary, change the first digit to zero, and convert back to decimal.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

Offset: 1

Henry Bottomley, Mar 22 2000

Triangle read by rows in which row n lists the first 2^n nonnegative integers (A001477), n >= 0. Right border gives A000225. Row sums give A006516. See example. - Omar E. Pol, Oct 17 2013

Without the initial zero also: zeroless numbers in base 3 (A032924: 1, 2, 11, 12, 21, ...), ternary digits decreased by 1 and read as binary. - M. F. Hasler, Jun 22 2020

			From _Omar E. Pol_, Oct 17 2013: (Start)
Written as an irregular triangle the sequence begins:
  0;
  0,1;
  0,1,2,3;
  0,1,2,3,4,5,6,7;
  0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15;
  ...
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, preprint, Theoretical Computer Sci., 98 (1992), 163-197.
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197 (see Ex. 24).
Index entries for sequences related to binary expansion of n

Cf. A000225, A000523, A002262, A004760, A006257, A006516, A030308, A036987, A053644, A062050, A083741, A160588.

a053645 1 = 0
a053645 n = 2 * a053645 n' + b  where (n', b) = divMod n 2
-- Reinhard Zumkeller, Aug 28 2014
a053645_list = concatMap (0 `enumFromTo`) a000225_list
-- Reinhard Zumkeller, Feb 04 2013, Mar 23 2012

[n - 2^Ilog2(n): n in [1..70]]; // Vincenzo Librandi, Jul 18 2019

seq(n - 2^ilog2(n), n=1..1000); # Robert Israel, Dec 23 2015

Table[n - 2^Floor[Log2[n]], {n, 100}] (* IWABUCHI Yu(u)ki, May 25 2017 *)
Table[FromDigits[Rest[IntegerDigits[n, 2]], 2], {n, 100}] (* IWABUCHI Yu(u)ki, May 25 2017 *)

a(n)=n-2^(#binary(n)-1) \\ Charles R Greathouse IV, Sep 02 2015

def a(n): return n - 2**(n.bit_length()-1)
print([a(n) for n in range(1, 85)]) # Michael S. Branicky, Jul 03 2021

def A053645(n): return n&(1<Chai Wah Wu, Jan 22 2023

a(n) = n - 2^A000523(n).
G.f.: 1/(1-x) * ((2x-1)/(1-x) + Sum_{k>=1} 2^(k-1)*x^2^k). - Ralf Stephan, Apr 18 2003
a(n) = (A006257(n)-1)/2. - N. J. A. Sloane, May 16 2003
a(1) = 0, a(2n) = 2a(n), a(2n+1) = 2a(n) + 1. - N. J. A. Sloane, Sep 13 2003
a(n) = A062050(n) - 1. - N. J. A. Sloane, Jun 12 2004
a(A004760(n+1)) = n. - Reinhard Zumkeller, May 20 2009
a(n) = f(n-1,1) with f(n,m) = if n < m then n else f(n-m,2*m). - Reinhard Zumkeller, May 20 2009
Conjecture: a(n) = (1 - A036987(n-1))*(1 + a(n-1)) for n > 1 with a(1) = 0. - Mikhail Kurkov, Jul 16 2019

A079000 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is odd".

Original entry on oeis.org

1, 4, 6, 7, 8, 9, 11, 13, 15, 16, 17, 18, 19, 20, 21, 23, 25, 27, 29, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 95, 97

Offset: 1

Matthew Vandermast, Feb 01 2003

a(a(n)) = 2n + 3 for n>1.

			a(2) cannot be 2 because 2 is even; it cannot be 3 because that would require 2 to be a member of the sequence. Hence a(2)=4 and the next odd member of the sequence is the fourth member.

Hsien-Kuei Hwang, S Janson, TH Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint, 2016; http://140.109.74.92/hk/wp-content/files/2016/12/aat-hhrr-1.pdf. Also Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585
N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.
N. J. A. Sloane, Seven Staggering Sequences.
Index entries for sequences of the a(a(n)) = 2n family

Cf. A079250-A079259, A079313, A079325, A064437, A003605, A079352, A079358.
Cf. also A080596, A080731, A080752, A121053, A080032.
Partial sums give A080566. Differences give A079948.

Digits := 50; A079000 := proc(n) local k,j; if n<=2 then n^2; else k := floor(evalf(log( (n+3)/6 )/log(2)) ); j := n-(9*2^k-3); 12*2^k-3+3*j/2 +abs(j)/2; fi; end;
A002264 := n->floor(n/3): A079944 := n->floor(log[2](4*(n+2)/3))-floor(log[2](n+2)): A000523 := n->floor(log[2](n)): f := n->A079944(A002264(n-4)): g := n->A000523(A002264(n+2)/2): A079000 := proc(n) if n>3 then RETURN(simplify(3*n+3-3*2^g(n)+(-1)^f(n)*(9*2^g(n)-n-3))/2) else if n>0 then RETURN([1,4,6][n]) else RETURN(0) fi fi: end;

a[1] = 1; a[n_] := (k = Floor[Log[2, (n+3)/6]]; j = n-(9*2^k - 3); 12*2^k-3 + 3*j/2 + Abs[j]/2); Table[a[n], {n, 1, 71}] (* Jean-François Alcover, May 21 2012, after Maple *)

a(1) = 1, a(2) = 4, then a(9*2^k-3+j) = 12*2^k-3+3*j/2+|j|/2 for k>=0, -3*2^k <= j <= 3*2^k. Also a(3n) = 3*b(n/3), a(3n+1) = 2*b(n)+b(n+1), a(3n+2) = b(n)+2*b(n+1) for n>=2, where b = A079905. - N. J. A. Sloane and Benoit Cloitre, Feb 20 2003
a(n+1) - 2*a(n) + a(n-1) = 1 for n = 9*2^k - 3, k>=0, = -1 for n = 2 and 3*2^k-3, k>=1 and = 0 otherwise.
a(n) = (3*n + 3 - 3*2^g(n) + (-1)^f(n)*(9*2^g(n) - n - 3))/2 for n>3, f(n) = A079944(A002264(n-4)) and g(n) = A000523(A002264(n+2)/2). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 23 2003
Also a(n) = n + 3*2^A000523(A002264(n+2)/2)*(1 - 3*A080584(n-4)) + A080584(n-4)*(n+3) for n>3, where A080584(n)=A079944(A002264(n)). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 24 2003

A007931 Numbers that contain only 1's and 2's. Nonempty binary strings of length n in lexicographic order.

Original entry on oeis.org

1, 2, 11, 12, 21, 22, 111, 112, 121, 122, 211, 212, 221, 222, 1111, 1112, 1121, 1122, 1211, 1212, 1221, 1222, 2111, 2112, 2121, 2122, 2211, 2212, 2221, 2222, 11111, 11112, 11121, 11122, 11211, 11212, 11221, 11222, 12111, 12112, 12121, 12122

Offset: 1

R. Muller

Numbers written in the dyadic system [Smullyan, Stillwell]. - N. J. A. Sloane, Feb 13 2019
Logic-binary sequence: prefix it by the empty word to have all binary words on the alphabet {1,2}.
The least binary word of length k is a(2^k - 1).
See Mathematica program for logic-binary sequence using (0,1) in place of (1,2); the sequence starts with 0,1,00,01,10. - Clark Kimberling, Feb 09 2012
A007953(a(n)) = A014701(n+1); A007954(a(n)) = A048896(n). - Reinhard Zumkeller, Oct 26 2012
a(n) is n written in base 2 where zeros are not allowed but twos are. The two distinct digits used are 1, 2 instead of 0, 1. To obtain this sequence from the "canonical" base 2 sequence with zeros allowed, just replace any 0 with a 2 and then subtract one from the group of digits situated on the left: (10-->2; 100-->12; 110-->22; 1000-->112; 1010-->122). - Robin Garcia, Jan 31 2014
For numbers made of only two different digits, see also A007088 (digits 0 & 1), A032810 (digits 2 & 3), A032834 (digits 3 & 4), A256290 (digits 4 & 5), A256291 (digits 5 & 6), A256292 (digits 6 & 7), A256340(digits 7 & 8), A256341 (digits 8 & 9), and A032804-A032816 (in other bases). Numbers with exactly two distinct (but unspecified) digits in base 10 are listed in A031955, for other bases in A031948-A031954. - M. F. Hasler, Apr 04 2015
The variant with digits {0, 1} instead of {1, 2} is obtained by deleting all initial digits in sequence A007088 (numbers written in base 2). - M. F. Hasler, Nov 03 2020

			Positive numbers may not start with 0 in the OEIS, otherwise this sequence would have been written as: 0, 1, 00, 01, 10, 11, 000, 001, 010, 011, 100, 101, 110, 111, 0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 00000, 00001, 00010, 00011, 00100, 00101, 00110, 00111, 01000, 01001, 01010, 01011, ...
From _Hieronymus Fischer_, Jun 06 2012: (Start)
a(10)   = 122.
a(100)  = 211212.
a(10^3) = 222212112.
a(10^4) = 1122211121112.
a(10^5) = 2111122121211112.
a(10^6) = 2221211112112111112.
a(10^7) = 11221112112122121111112.
a(10^8) = 12222212122221111211111112.
a(10^9) = 22122211221212211212111111112. (End)

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 2. - From N. J. A. Sloane, Jul 26 2012
K. Atanassov, On the 97th, 98th and the 99th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 3, 89-93.
R. M. Smullyan, Theory of Formal Systems, Princeton, 1961.
John Stillwell, Reverse Mathematics, Princeton, 2018. See p. 90.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000 (terms up to 2^10-2 from T. D. Noe, corrected by Sean A. Irvine, April 18 2019)
K. Atanassov, On Some of Smarandache's Problems, American Research Press, 1999, 16-21.
EMIS, Mirror site for Southwest Journal of Pure and Applied Mathematics
R. R. Forslund, A logical alternative to the existing positional number system, Southwest Journal of Pure and Applied Mathematics, Vol. 1, 1995.
R. R. Forslund, Positive Integer Pages
James E. Foster, A Number System without a Zero-Symbol, Mathematics Magazine, Vol. 21, No. 1. (1947), pp. 39-41.
F. Smarandache, Only Problems, Not Solutions!.
Index entries for 10-automatic sequences.

Cf. A007932 (digits 1-3), A059893, A045670, A052382 (digits 1-9), A059939, A059941, A059943, A032924, A084544, A084545, A046034 (prime digits 2,3,5,7), A089581, A084984 (no prime digits); A001742, A001743, A001744: loops; A202267 (digits 0, 1 and primes), A202268 (digits 1,4,6,8,9), A014261 (odd digits), A014263 (even digits).
Cf. A007088 (digits 0 & 1), A032810 (digits 2 & 3), A032834 (digits 3 & 4), A256290 (digits 4 & 5), A256291 (digits 5 & 6), A256292 (digits 6 & 7), A256340 (digits 7 & 8), A256341 (digits 8 & 9), and A032804-A032816 (in other bases).
Cf. A020450 (primes).

a007931 n = f (n + 1) where
   f x = if x < 2 then 0 else (10 * f x') + m + 1
     where (x', m) = divMod x 2
-- Reinhard Zumkeller, Oct 26 2012

[n: n in [1..100000] | Set(Intseq(n)) subset {1,2}]; // Vincenzo Librandi, Aug 19 2016

# Maple program to produce the sequence:
a:= proc(n) local m, r, d; m, r:= n, 0;
      while m>0 do d:= irem(m, 2, 'm');
        if d=0 then d:=2; m:= m-1 fi;
        r:= d, r
      od; parse(cat(r))/10
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 26 2016
# Maple program to invert this sequence: given a(n), it returns n. - N. J. A. Sloane, Jul 09 2012
invert7931:=proc(u)
local t1,t2,i;
t1:=convert(u,base,10);
[seq(t1[i]-1,i=1..nops(t1))];
[op(%),1];
t2:=convert(%,base,2,10);
add(t2[i]*10^(i-1),i=1..nops(t2))-1;
end;

f[n_] := FromDigits[Rest@IntegerDigits[n + 1, 2] + 1]; Array[f, 42] (* Robert G. Wilson v Sep 14 2006 *)
(* Next, A007931 using (0,1) instead of (1,2) *)
d[n_] := FromDigits[Rest@IntegerDigits[n + 1, 2] + 1]; Array[FromCharacterCode[ToCharacterCode[ToString[d[#]]] - 1] &, 100] (* Peter J. C. Moses, at request of Clark Kimberling, Feb 09 2012 *)
Flatten[Table[FromDigits/@Tuples[{1,2},n],{n,5}]] (* Harvey P. Dale, Sep 13 2014 *)

apply( {A007931(n)=fromdigits([d+1|d<-binary(n+1)[^1]])}, [1..44]) \\ M. F. Hasler, Nov 03 2020, replacing older code from Mar 26 2015

/* inverse function */ apply( {A007931_inv(N)=fromdigits([d-1|d<-digits(N)],2)+2<M. F. Hasler, Nov 09 2020

def a(n): return int(bin(n+1)[3:].replace('1', '2').replace('0', '1'))
print([a(n) for n in range(1, 45)]) # Michael S. Branicky, May 13 2021

def A007931(n): return int(s:=bin(n+1)[3:])+(10**(len(s))-1)//9 # Chai Wah Wu, Jun 13 2025

To get a(n), write n+1 in base 2, remove initial 1, add 1 to all remaining digits: e.g., eleven (11) in base 2 is 1011; remove initial 1 and add 1 to remaining digits: a(10)=122. - Clark Kimberling, Mar 11 2003
Conversely, given a(n), to get n: subtract 1 from all digits, prefix with an initial 1, convert this binary number to base 10, subtract 1. E.g., a(6)=22 -> 11 -> 111 -> 7 -> 6. - N. J. A. Sloane, Jul 09 2012
a(n) = A053645(n+1)+A002275(A000523(n)) = a(n-2^b(n))+10^b(n) where b(n) = A059939(n) = floor(log_2(n+1)-1). - Henry Bottomley, Feb 14 2001
From Hieronymus Fischer, Jun 06 2012 and Jun 08 2012: (Start)
The formulas are designed to calculate base-10 numbers only using the digits 1 and 2.
a(n) = Sum_{j=0..m-1} (1 + b(j) mod 2)*10^j, where m = floor(log_2(n+1)), b(j) = floor((n+1-2^m)/(2^j)).
Special values:
a(k*(2^n-1)) = k*(10^n-1)/9, k= 1,2.
a(3*2^n-2) = (11*10^n-2)/9 = 10^n+2*(10^n-1)/9.
a(2^n-2) = 2*(10^(n-1)-1)/9, n>1.
Inequalities:
a(n) <= (10^log_2(n+1)-1)/9, equality holds for n=2^k-1, k>0.
a(n) > (2/10)*(10^log_2(n+1)-1)/9.
Lower and upper limits:
lim inf a(n)/10^log_2(n) = 1/45, for n --> infinity.
lim sup a(n)/10^log_2(n) = 1/9, for n --> infinity.
G.f.: g(x) = (1/(x(1-x)))*sum_{j=0..infinity} 10^j* x^(2*2^j)*(1 + 2 x^2^j)/(1 + x^2^j).
Also: g(x) = (1/(1-x))*(h_(2,0)(x) + h_(2,1)(x) - 2*h_(2,2)(x)), where h_(2,k)(x) = sum_{j>=0} 10^j*x^(2^(j+1)-1)*x^(k*2^j)/(1-x^2^(j+1)).
Also: g(x) = (1/(1-x)) sum_{j>=0} (1 - 3(x^2^j)^2 + 2(x^2^j)^3)*x^2^j*f_j(x)/(1-x^2^j), where f_j(x) = 10^j*x^(2^j-1)/(1-(x^2^j)^2). The f_j obey the recurrence f_0(x) = 1/(1-x^2), f_(j+1)(x) = 10x*f_j(x^2). (End)

Some crossrefs added by Hieronymus Fischer, Jun 06 2012

Edited by M. F. Hasler, Mar 26 2015

A023758 Numbers of the form 2^i - 2^j with i >= j.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 12, 14, 15, 16, 24, 28, 30, 31, 32, 48, 56, 60, 62, 63, 64, 96, 112, 120, 124, 126, 127, 128, 192, 224, 240, 248, 252, 254, 255, 256, 384, 448, 480, 496, 504, 508, 510, 511, 512, 768, 896, 960, 992, 1008, 1016, 1020, 1022, 1023

Offset: 1

Olivier Gérard

Numbers whose digits in base 2 are in nonincreasing order.
Might be called "nialpdromes".
Subset of A077436. Proof: Since a(n) is of the form (2^i-1)*2^j, i,j >= 0, a(n)^2 = (2^(2i) - 2^(i+1))*2^(2j) + 2^(2j) where the first sum term has i-1 one bits and its 2j-th bit is zero, while the second sum term switches the 2j-th bit to one, giving i one bits, as in a(n). - Ralf Stephan, Mar 08 2004
Numbers whose binary representation contains no "01". - Benoit Cloitre, May 23 2004
Every polynomial with coefficients equal to 1 for the leading terms and 0 after that, evaluated at 2. For instance a(13) = x^4 + x^3 + x^2 at 2, a(14) = x^4 + x^3 + x^2 + x at 2. - Ben Paul Thurston, Jan 11 2008
From Gary W. Adamson, Jul 18 2008: (Start)
As a triangle by rows starting:
   1;
   2,  3;
   4,  6,  7;
   8, 12, 14, 15;
  16, 24, 28, 30, 31;
  ...,
equals A000012 * A130123 * A000012, where A130123 = (1, 0,2; 0,0,4; 0,0,0,8; ...). Row sums of this triangle = A000337 starting (1, 5, 17, 49, 129, ...). (End)
First differences are A057728 = 1; 1; 1; 1; 2,1; 1; 4,2,1; 1; 8,4,2,1; 1; ... i.e., decreasing powers of 2, separated by another "1". - M. F. Hasler, May 06 2009
Apart from first term, numbers that are powers of 2 or the sum of some consecutive powers of 2. - Omar E. Pol, Feb 14 2013
From Andres Cicuttin, Apr 29 2016: (Start)
Numbers that can be digitally generated with twisted ring (Johnson) counters. This is, the binary digits of a(n) correspond to those stored in a shift register where the input bit of the first bit storage element is the inverted output of the last storage element. After starting with all 0’s, each new state is obtained by rotating the stored bits but inverting at each state transition the last bit that goes to the first position (see link).
Examples: for a(n) represented by three bits
             Binary
a(5)= 4   ->  100   last bit = 0
a(6)= 6   ->  110   first bit = 1 (inverted last bit of previous number)
a(7)= 7   ->  111
and for a(n) represented by four bits
             Binary
a(8) = 8   -> 1000
a(9) = 12  -> 1100  last bit = 0
a(10)= 14  -> 1110  first bit = 1 (inverted last bit of previous number)
a(11)= 15  -> 1111
(End)
Powers of 2 represented in bases which are terms of this sequence must always contain at least one digit which is also a power of 2. This is because 2^i mod (2^i - 2^j) = 2^j, which means the last digit always cycles through powers of 2 (or if i=j+1 then the first digit is a power of 2 and the rest are trailing zeros). The only known non-member of this sequence with this property is 5. - Ely Golden, Sep 05 2017
Numbers k such that k = 2^(1 + A000523(k)) - 2^A007814(k). - Daniel Starodubtsev, Aug 05 2021
A002260(n) = v(a(n)/2^v(a(n))+1) and A002024(n) = A002260(n) + v(a(n)) where v is the dyadic valuation (i.e., A007814). - Lorenzo Sauras Altuzarra, Feb 01 2023

			a(22) = 64 = 32 + 32 = 2^5 + a(16) = 2^A003056(20) + a(22-5-1).
a(23) = 96 = 64 + 32 = 2^6 + a(16) = 2^A003056(21) + a(23-6-1).
a(24) = 112 = 64 + 48 = 2^6 + a(17) = 2^A003056(22) + a(24-6-1).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 5051 terms from T. D. Noe)
Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
S. M. Shabab Hossain, Md. Mahmudur Rahman and M. Sohel Rahman, Solving a Generalized Version of the Exact Cover Problem with a Light-Based Device, Optical Supercomputing, Lecture Notes in Computer Science, 2011, Volume 6748/2011, 23-31, DOI: 10.1007/978-3-642-22494-2_4.
Eric Weisstein's World of Mathematics, Digit.
Wikipedia, Ring counter.
Index entries for 2-automatic sequences.
Index entries for sequences related to binary expansion of n.

A000337(r) = sum of row T(r, c) with 0 <= c < r. See also A002024, A003056, A140129, A140130, A221975.
Cf. A007088, A130123, A101082 (complement), A340375 (characteristic function).
This is the base-2 version of A064222. First differences are A057728.
Subsequence of A077436, of A129523, of A277704, and of A333762.
Subsequences: A043569 (nonzero even terms, or equally, nonzero terms doubled), A175332, A272615, A335431, A000396 (its even terms only), A324200.
Positions of zeros in A049502, A265397, A277899, A284264.
Positions of ones in A283983, A283989.
Positions of nonzero terms in A341509 (apart from the initial zero).
Positions of squarefree terms in A260443.
Fixed points of A264977, A277711, A283165, A334666.
Distinct terms in A340632.
Cf. also A002024, A002260, A007814, A133797, A152449, A181666, A211705, A277699, A306441.
Cf. also A309758, A309759, A309761 (for analogous sequences).

import Data.Set (singleton, deleteFindMin, insert)
a023758 n = a023758_list !! (n-1)
a023758_list = 0 : f (singleton 1) where
f s = x : f (if even x then insert z s' else insert z $ insert (z+1) s')
where z = 2*x; (x, s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 24 2014, Dec 19 2012

a:=proc(n) local n2,d: n2:=convert(n,base,2): d:={seq(n2[j]-n2[j-1],j=2..nops(n2))}: if n=0 then 0 elif n=1 then 1 elif d={0,1} or d={0} or d={1} then n else fi end: seq(a(n),n=0..2100); # Emeric Deutsch, Apr 22 2006

Union[Flatten[Table[2^i - 2^j, {i, 0, 100}, {j, 0, i}]]] (* T. D. Noe, Mar 15 2011 *)
Select[Range[0, 2^10], NoneTrue[Differences@ IntegerDigits[#, 2], # > 0 &] &] (* Michael De Vlieger, Sep 05 2017 *)

for(n=0,2500,if(prod(k=1,length(binary(n))-1,component(binary(n),k)+1-component(binary(n),k+1))>0,print1(n,",")))

A023758(n)= my(r=round(sqrt(2*n--))); (1<<(n-r*(r-1)/2)-1)<<(r*(r+1)/2-n)
/* or, to illustrate the "decreasing digit" property and analogy to A064222: */
A023758(n,show=0)={ my(a=0); while(n--, show & print1(a","); a=vecsort(binary(a+1)); a*=vector(#a,j,2^(j-1))~); a} \\ M. F. Hasler, May 06 2009

is(n)=if(n<5,1,n>>=valuation(n,2);n++;n>>valuation(n,2)==1) \\ Charles R Greathouse IV, Jan 04 2016

list(lim)=my(v=List([0]),t); for(i=1,logint(lim\1+1,2), t=2^i-1; while(t<=lim, listput(v,t); t*=2)); Set(v) \\ Charles R Greathouse IV, May 03 2016

def a_next(a_n): return (a_n | (a_n >> 1)) + (a_n & 1)
a_n = 1; a = [0]
for i in range(55): a.append(a_n); a_n = a_next(a_n) # Falk Hüffner, Feb 19 2022

from math import isqrt
def A023758(n): return (1<<(m:=isqrt(n-1<<3)+1>>1))-(1<<(m*(m+1)-(n-1<<1)>>1)) # Chai Wah Wu, Feb 23 2025

a(n) = 2^s(n) - 2^((s(n)^2 + s(n) - 2n)/2) where s(n) = ceiling((-1 + sqrt(1+8n))/2). - Sam Alexander, Jan 08 2005
a(n) = 2^k + a(n-k-1) for 1 < n and k = A003056(n-2). The rows of T(r, c) = 2^r-2^c for 0 <= c < r read from right to left produce this sequence: 1; 2, 3; 4, 6, 7; 8, 12, 14, 15; ... - Frank Ellermann, Dec 06 2001
For n > 0, a(n) mod 2 = A010054(n). - Benoit Cloitre, May 23 2004
A140130(a(n)) = 1 and for n > 1: A140129(a(n)) = A002262(n-2). - Reinhard Zumkeller, May 14 2008
a(n+1) = (2^(n - r(r-1)/2) - 1) 2^(r(r+1)/2 - n), where r=round(sqrt(2n)). - M. F. Hasler, May 06 2009
Start with A000225. If k is in the sequence, then so is 2k. - Ralf Stephan, Aug 16 2013
G.f.: (x^2/((2-x)*(1-x)))*(1 + Sum_{k>=0} x^((k^2+k)/2)*(1 + x*(2^k-1))). The sum is related to Jacobi theta functions. - Robert Israel, Feb 24 2015
A049502(a(n)) = 0. - Reinhard Zumkeller, Jun 17 2015
a(n) = a(n-1) + a(n-d)/a(d*(d+1)/2 + 2) if n > 1, d > 0, where d = A002262(n-2). - Yuchun Ji, May 11 2020
A277699(a(n)) = a(n)^2, A306441(a(n)) = a(n+1). - Antti Karttunen, Feb 15 2021 (the latter identity from A306441)
Sum_{n>=2} 1/a(n) = A211705. - Amiram Eldar, Feb 20 2022

Definition changed by N. J. A. Sloane, Jan 05 2008

A065120 Highest power of 2 dividing A057335(n).

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 1, 1, 4, 3, 2, 2, 1, 1, 1, 1, 5, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 6, 5, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 6, 5, 5, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Alford Arnold, Nov 12 2001

a(n) appears on row 1 of the array illustrated in A066099.
Except for initial zero, ordinal transform of A062050. After initial zero, n-th chunk consists of n, one n-1, two (n-2)'s, ..., 2^(k-1) (n-k)'s, ..., 2^(n-1) 1's. - Franklin T. Adams-Watters, Sep 11 2006
Zero together with a triangle read by rows in which row j lists the first 2^(j-1) terms of A001511 in nonincreasing order, j >= 1, see example. Also row j lists the first parts, in nonincreasing order, of the compositions of j. - Omar E. Pol, Sep 11 2013
The n-th row represents the frequency distribution of 1, 2, 3, ..., 2^(n-1) in the first 2^n - 1 terms of A003602. - Gary W. Adamson, Jun 10 2021

			A057335(7)= 30 and 30 = 2*3*5 so a(7) = 1; A057335(9)= 24 and 24 = 8*3 so a(9) = 3
From _Omar E. Pol_, Aug 30 2013: (Start)
Written as an irregular triangle with row lengths A011782:
  0;
  1;
  2,1;
  3,2,1,1;
  4,3,2,2,1,1,1,1;
  5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1;
  6,5,4,4,3,3,3,3,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;
  ...
Column 1 is A001477. Row sums give A000225. Row lengths is A011782.
(End)

Cf. A066099, A062050, A011782.

Cf. A003602.

nmax = 105;
A062050 = Flatten[Table[Range[2^n], {n, 0, Log[2, nmax] // Ceiling}]];
Module[{b}, b[_] = 0;
a[n_] := If[n == 0, 0, With[{t = A062050[[n]]}, b[t] = b[t] + 1]]];
a /@ Range[0, nmax] (* Jean-François Alcover, Jan 12 2022 *)

lista(nn) = {my(v = vector(nn)); v[1] = 1; for (i=2, nn, v[i] = mg(i-1)*v[(i+1)\2];); for (i=1, nn, print1(valuation(v[i], 2), ", "););} \\ Michel Marcus, Feb 09 2014

my(L(n)=if(n,logint(n,2),-1)); a(n) = my(p=L(n)); p - L(n-1<Kevin Ryde, Aug 06 2021

From Daniel Starodubtsev, Aug 05 2021: (Start)
a(n) = A001511(A059894(n) - 2^A000523(n) + 1) for n > 0 with a(0) = 0.
a(2n+1) = a(n), a(2n) = a(n) + A036987(n-1) for n > 1 with a(0) = 0, a(1) = 1. (End)

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003

cp's OEIS Frontend

A036987 Fredholm-Rueppel sequence.

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A053644 Most significant bit of n, msb(n); largest power of 2 less than or equal to n; write n in binary and change all but the first digit to zero.

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A059893 Reverse the order of all but the most significant bit in binary expansion of n: if n = 1ab..yz then a(n) = 1zy..ba.

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A053645 Distance to largest power of 2 less than or equal to n; write n in binary, change the first digit to zero, and convert back to decimal.

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A079000 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is odd".

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