A003986 -id:A003986 - cp's OEIS frontend

A007814 Exponent of highest power of 2 dividing n, a.k.a. the binary carry sequence, the ruler sequence, or the 2-adic valuation of n.

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

Offset: 1

John Tromp, Dec 11 1996

This sequence is an exception to my usual rule that when every other term of a sequence is 0 then those 0's should be omitted. In this case we would get A001511. - N. J. A. Sloane
To construct the sequence: start with 0,1, concatenate to get 0,1,0,1. Add + 1 to last term gives 0,1,0,2. Concatenate those 4 terms to get 0,1,0,2,0,1,0,2. Add + 1 to last term etc. - Benoit Cloitre, Mar 06 2003
The sequence is invariant under the following two transformations: increment every element by one (1, 2, 1, 3, 1, 2, 1, 4, ...), put a zero in front and between adjacent elements (0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, ...). The intermediate result is A001511. - Ralf Hinze (ralf(AT)informatik.uni-bonn.de), Aug 26 2003
Fixed point of the morphism 0->01, 1->02, 2->03, 3->04, ..., n->0(n+1), ..., starting from a(1) = 0. - Philippe Deléham, Mar 15 2004
Fixed point of the morphism 0->010, 1->2, 2->3, ..., n->(n+1), .... - Joerg Arndt, Apr 29 2014
a(n) is also the number of times to repeat a step on an even number in the hailstone sequence referenced in the Collatz conjecture. - Alex T. Flood (whiteangelsgrace(AT)gmail.com), Sep 22 2006
Let F(n) be the n-th Fermat number (A000215). Then F(a(r-1)) divides F(n)+2^k for r = k mod 2^n and r != 1. - T. D. Noe, Jul 12 2007
The following relation holds: 2^A007814(n)*(2*A025480(n-1)+1) = A001477(n) = n. (See functions hd, tl and cons in [Paul Tarau 2009].)
a(n) is the number of 0's at the end of n when n is written in base 2.
a(n+1) is the number of 1's at the end of n when n is written in base 2. - M. F. Hasler, Aug 25 2012
Shows which bit to flip when creating the binary reflected Gray code (bits are numbered from the right, offset is 0). That is, A003188(n) XOR A003188(n+1) == 2^A007814(n). - Russ Cox, Dec 04 2010
The sequence is squarefree (in the sense of not containing any subsequence of the form XX) [Allouche and Shallit]. Of course it contains individual terms that are squares (such as 4). - Comment expanded by N. J. A. Sloane, Jan 28 2019
a(n) is the number of zero coefficients in the n-th Stern polynomial, A125184. - T. D. Noe, Mar 01 2011
Lemma: For n < m with r = a(n) = a(m) there exists n < k < m with a(k) > r. Proof: We have n=b2^r and m=c2^r with b < c both odd; choose an even i between them; now a(i2^r) > r and n < i2^r < m. QED. Corollary: Every finite run of consecutive integers has a unique maximum 2-adic valuation. - Jason Kimberley, Sep 09 2011
a(n-2) is the 2-adic valuation of A000166(n) for n >= 2. - Joerg Arndt, Sep 06 2014
a(n) = number of 1's in the partition having Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product_{j=1..r} p_j-th prime (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: a(24)=3; indeed, the partition having Heinz number 24 = 2*2*2*3  is [1,1,1,2]. - Emeric Deutsch, Jun 04 2015
a(n+1) is the difference between the two largest parts in the integer partition having viabin number n (0 is assumed to be a part). Example: a(20) = 2. Indeed, we have 19 = 10011_2, leading to the Ferrers board of the partition [3,1,1]. For the definition of viabin number see the comment in A290253. - Emeric Deutsch, Aug 24 2017
Apart from being squarefree, as noted above, the sequence has the property that every consecutive subsequence contains at least one number an odd number of times. - Jon Richfield, Dec 20 2018
a(n+1) is the 2-adic valuation of Sum_{e=0..n} u^e = (1 + u + u^2 + ... + u^n), for any u of the form 4k+1 (A016813). - Antti Karttunen, Aug 15 2020
{a(n)} represents the "first black hat" strategy for the game of countably infinitely many hats, with a probability of success of 1/3; cf. the Numberphile link below. - Frederic Ruget, Jun 14 2021
a(n) is the least nonnegative integer k for which there does not exist i+j=n and a(i)=a(j)=k (cf. A322523). - Rémy Sigrist and Jianing Song, Aug 23 2022

			2^3 divides 24, so a(24)=3.
From _Omar E. Pol_, Jun 12 2009: (Start)
Triangle begins:
  0;
  1,0;
  2,0,1,0;
  3,0,1,0,2,0,1,0;
  4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0;
  5,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0;
  6,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,5,0,1,0,2,...
(End)

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 27.
K. Atanassov, On the 37th and the 38th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 2, 83-85.
Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.

T. D. Noe, Table of n, a(n) for n = 1..10000
Joerg Arndt, Subset-lex: did we miss an order?, arXiv:1405.6503 [math.CO], 2014.
K. Atanassov, On Some of Smarandache's Problems
Alain Connes, Caterina Consani, and Henri Moscovici, Zeta zeros and prolate wave operators, arXiv:2310.18423 [math.NT], 2023.
Dario T. de Castro, P-adic Order of Positive Integers via Binomial Coefficients, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 22, Paper A61, 2022.
Mathieu Guay-Paquet and Jeffrey Shallit, Avoiding Squares and Overlaps Over the Natural Numbers, (2009) Discrete Math., 309 (2009), 6245-6254.
Mathieu Guay-Paquet and Jeffrey Shallit, Avoiding Squares and Overlaps Over the Natural Numbers, arXiv:0901.1397 [math.CO], 2009.
M. Hassani, Equations and inequalities involving v_p(n!), J. Inequ. Pure Appl. Math. 6 (2005) vol. 2, #29.
A. M. Hinz, S. Klavžar, U. Milutinović, and C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 61. Book's website
R. Hinze, Concrete stream calculus: An extended study, J. Funct. Progr. 20 (5-6) (2010) 463-535, doi, Section 3.2.3.
Clark Kimberling, Affinely recursive sets and orderings of languages, Discrete Math., 274 (2004), 147-160.
Francis Laclé, 2-adic parity explorations of the 3n+ 1 problem, hal-03201180v2 [cs.DM], 2021.
Shuo Li, Palindromic length sequence of the ruler sequence and of the period-doubling sequence, arXiv:2007.08317 [math.CO], 2020.
Nicolas Mallet, Trial for a proof of the Syracuse conjecture, arXiv preprint arXiv:1507.05039 [math.GM], 2015.
S. Mazzanti, Plain Bases for Classes of Primitive Recursive Functions, Mathematical Logic Quarterly, 48 (2002).
Matthew Andres Moreno, Luis Zaman, and Emily Dolson, Structured Downsampling for Fast, Memory-efficient Curation of Online Data Streams, arXiv:2409.06199 [cs.DS], 2024. See p. 5.
Sascha Mücke, Coding Nuggets Faster QUBO Brute-Force Solving, TU Dortmund Univ. (Germany 2023).
S. Northshield, An Analogue of Stern's Sequence for Z[sqrt(2)], Journal of Integer Sequences, 18 (2015), #15.11.6.
Numberphile, Hat Problems - Numberphile
Giovanni Pighizzini, Limited Automata: Properties, Complexity and Variants, International Conference on Descriptional Complexity of Formal Systems (DCFS 2019) Descriptional Complexity of Formal Systems, Lecture Notes in Computer Science (LNCS, Vol. 11612) Springer, Cham, 57-73.
Simon Plouffe, On the values of the functions ... [zeta and Gamma] ..., arXiv preprint arXiv:1310.7195 [math.NT], 2013.
A. Postnikov (MIT) and B. Sagan, What power of two divides a weighted Catalan number?, arXiv:math/0601339 [math.CO], 2006.
Lara Pudwell and Eric Rowland, Avoiding fractional powers over the natural numbers, arXiv:1510.02807 [math.CO] (2015). Also Electronic Journal of Combinatorics, Volume 25(2) (2018), #P2.27. See Section 2.
Ville Salo, Decidability and Universality of Quasiminimal Subshifts, arXiv preprint arXiv:1411.6644 [math.DS], 2014.
Vladimir Shevelev, Several results on sequences which are similar to the positive integers, arXiv:0904.2101 [math.NT], 2014.
F. Smarandache, Only Problems, Not Solutions!.
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Paul Tarau, A Groupoid of Isomorphic Data Transformations. Calculemus 2009, 8th International Conference, MKM 2009, pp. 170-185, Springer, LNAI 5625.
P. M. B. Vitanyi, An optimal simulation of counter machines, SIAM J. Comput, 14:1(1985), 1-33.
Eric Weisstein's World of Mathematics, Binary, Binary Carry Sequence, and Double-Free Set.
Wikipedia, P-adic order.
Index entries for sequences that are fixed points of mappings
Index entries for sequences related to binary expansion of n

Cf. A011371 (partial sums), A094267 (first differences), A001511 (bisection), A346070 (mod 4).
Bisection of A050605 and |A088705|. Pairwise sums are A050603 and A136480. Difference of A285406 and A281264.
This is Guy Steele's sequence GS(1, 4) (see A135416). Cf. A053398(1,n). Column/row 1 of table A050602.
Cf. A007949 (3-adic), A235127 (4-adic), A112765 (5-adic), A122841 (6-adic), A214411 (7-adic), A244413 (8-adic), A122840 (10-adic).
Cf. A000079, A002487, A006519, A051064, A055457, A063787, A215366, A220466.
Cf. A086463 (Dgf at s=2).

a007814 n = if m == 0 then 1 + a007814 n' else 0
            where (n', m) = divMod n 2
-- Reinhard Zumkeller, Jul 05 2013, May 14 2011, Apr 08 2011

a007814 n | odd n = 0 | otherwise = 1 + a007814 (n `div` 2)
--  Walt Rorie-Baety, Mar 22 2013

[Valuation(n, 2): n in [1..120]]; // Bruno Berselli, Aug 05 2013

ord := proc(n) local i,j; if n=0 then return 0; fi; i:=0; j:=n; while j mod 2 <> 1 do i:=i+1; j:=j/2; od: i; end proc: seq(ord(n), n=1..111);
A007814 := n -> padic[ordp](n,2): seq(A007814(n), n=1..111); # Peter Luschny, Nov 26 2010

Table[IntegerExponent[n, 2], {n, 64}] (* Eric W. Weisstein *)
IntegerExponent[Range[64], 2] (* Eric W. Weisstein, Feb 01 2024 *)
p=2; Array[ If[ Mod[ #, p ]==0, Select[ FactorInteger[ # ], Function[ q, q[ [ 1 ] ]==p ], 1 ][ [ 1, 2 ] ], 0 ]&, 96 ]
DigitCount[BitXor[x, x - 1], 2, 1] - 1; a different version based on the same concept: Floor[Log[2, BitXor[x, x - 1]]] (* Jaume Simon Gispert (jaume(AT)nuem.com), Aug 29 2004 *)
Nest[Join[ #, ReplacePart[ #, Length[ # ] -> Last[ # ] + 1]] &, {0, 1}, 5] (* N. J. Gunther, May 23 2009 *)
Nest[ Flatten[# /. a_Integer -> {0, a + 1}] &, {0}, 7] (* Robert G. Wilson v, Jan 17 2011 *)

PARI
```
A007814(n)=valuation(n,2);
```

import math
def a(n): return int(math.log(n - (n & n - 1), 2)) # Indranil Ghosh, Apr 18 2017

def A007814(n): return (~n & n-1).bit_length() # Chai Wah Wu, Jul 01 2022

sapply(1:100,function(x) sum(gmp::factorize(x)==2)) # Christian N. K. Anderson, Jun 20 2013

(define (A007814 n) (let loop ((n n) (e 0)) (if (odd? n) e (loop (/ n 2) (+ 1 e))))) ;; Antti Karttunen, Oct 06 2017

a(n) = A001511(n) - 1.
a(2*n) = A050603(2*n) = A001511(n).
a(n) = A091090(n-1) + A036987(n-1) - 1.
a(n) = 0 if n is odd, otherwise 1 + a(n/2). - Reinhard Zumkeller, Aug 11 2001
Sum_{k=1..n} a(k) = n - A000120(n). - Benoit Cloitre, Oct 19 2002
G.f.: A(x) = Sum_{k>=1} x^(2^k)/(1-x^(2^k)). - Ralf Stephan, Apr 10 2002
G.f. A(x) satisfies A(x) = A(x^2) + x^2/(1-x^2). A(x) = B(x^2) = B(x) - x/(1-x), where B(x) is the g.f. for A001151. - Franklin T. Adams-Watters, Feb 09 2006
Totally additive with a(p) = 1 if p = 2, 0 otherwise.
Dirichlet g.f.: zeta(s)/(2^s-1). - Ralf Stephan, Jun 17 2007
Define 0 <= k <= 2^n - 1; binary: k = b(0) + 2*b(1) + 4*b(2) + ... + 2^(n-1)*b(n-1); where b(x) are 0 or 1 for 0 <= x <= n - 1; define c(x) = 1 - b(x) for 0 <= x <= n - 1; Then: a(k) = c(0) + c(0)*c(1) + c(0)*c(1)*c(2) + ... + c(0)*c(1)...c(n-1); a(k+1) = b(0) + b(0)*b(1) + b(0)*b(1)*b(2) + ... + b(0)*b(1)...b(n-1). - Arie Werksma (werksma(AT)tiscali.nl), May 10 2008
a(n) = floor(A002487(n - 1) / A002487(n)). - Reikku Kulon, Oct 05 2008
Sum_{k=1..n} (-1)^A000120(n-k)*a(k) = (-1)^(A000120(n)-1)*(A000120(n) - A000035(n)). - Vladimir Shevelev, Mar 17 2009
a(A001147(n) + A057077(n-1)) = a(2*n). - Vladimir Shevelev, Mar 21 2009
For n>=1, a(A004760(n+1)) = a(n). - Vladimir Shevelev, Apr 15 2009
2^(a(n)) = A006519(n). - Philippe Deléham, Apr 22 2009
a(n) = A063787(n) - A000120(n). - Gary W. Adamson, Jun 04 2009
a(C(n,k)) = A000120(k) + A000120(n-k) - A000120(n). - Vladimir Shevelev, Jul 19 2009
a(n!) = n - A000120(n). - Vladimir Shevelev, Jul 20 2009
v_{2}(n) = Sum_{r>=1} (r / 2^(r+1)) Sum_{k=0..2^(r+1)-1} e^(2(k*Pi*i(n+2^r))/(2^(r+1))). - A. Neves, Sep 28 2010, corrected Oct 04 2010
a(n) mod 2 = A096268(n-1). - Robert G. Wilson v, Jan 18 2012
a(A005408(n)) = 1; a(A016825(n)) = 3; A017113(a(n)) = 5; A051062(a(n)) = 7; a(n) = (A037227(n)-1)/2. - Reinhard Zumkeller, Jun 30 2012
a((2*n-1)*2^p) = p, p >= 0 and n >= 1. - Johannes W. Meijer, Feb 04 2013
a(n) = A067255(n,1). - Reinhard Zumkeller, Jun 11 2013
a(n) = log_2(n - (n AND n-1)). - Gary Detlefs, Jun 13 2014
a(n) = 1 + A000120(n-1) - A000120(n), where A000120 is the Hamming weight function. - Stanislav Sykora, Jul 14 2014
A053398(n,k) = a(A003986(n-1,k-1)+1); a(n) = A053398(n,1) = A053398(n,n) = A053398(2*n-1,n) = Min_{k=1..n} A053398(n,k). - Reinhard Zumkeller, Aug 04 2014
a((2*x-1)*2^n) = a((2*y-1)*2^n) for positive n, x and y. - Juri-Stepan Gerasimov, Aug 04 2016
a(n) = A285406(n) - A281264(n). - Ralf Steiner, Apr 18 2017
a(n) = A000005(n)/(A000005(2*n) - A000005(n)) - 1. - conjectured by Velin Yanev, Jun 30 2017, proved by Nicholas Stearns, Sep 11 2017
Equivalently to above formula, a(n) = A183063(n) / A001227(n), i.e., a(n) is the number of even divisors of n divided by number of odd divisors of n. - Franklin T. Adams-Watters, Oct 31 2018
a(n)*(n mod 4) = 2*floor(((n+1) mod 4)/3). - Gary Detlefs, Feb 16 2019
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1. - Amiram Eldar, Jul 11 2020
a(n) = 2*Sum_{j=1..floor(log_2(n))} frac(binomial(n, 2^j)*2^(j-1)/n). - Dario T. de Castro, Jul 08 2022
a(n) = A070939(n) - A070939(A030101(n)). - Andrew T. Porter, Dec 16 2022
a(n) = floor((gcd(n, 2^n)^(n+1) mod (2^(n+1)-1)^2)/(2^(n+1)-1)) (see Lemma 3.4 from Mazzanti's 2002 article). - Lorenzo Sauras Altuzarra, Mar 10 2024
a(n) = 1 - A088705(n). - Chai Wah Wu, Sep 18 2024

Formula index adapted to the offset of A025480 by R. J. Mathar, Jul 20 2010

Edited by Ralf Stephan, Feb 08 2014

A003987 Table of n XOR m (or Nim-sum of n and m) read by antidiagonals with m>=0, n>=0.

Original entry on oeis.org

0, 1, 1, 2, 0, 2, 3, 3, 3, 3, 4, 2, 0, 2, 4, 5, 5, 1, 1, 5, 5, 6, 4, 6, 0, 6, 4, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 6, 4, 6, 0, 6, 4, 6, 8, 9, 9, 5, 5, 1, 1, 5, 5, 9, 9, 10, 8, 10, 4, 2, 0, 2, 4, 10, 8, 10, 11, 11, 11, 11, 3, 3, 3, 3, 11, 11, 11, 11, 12, 10, 8, 10, 12, 2, 0, 2, 12, 10, 8, 10, 12, 13, 13, 9, 9, 13, 13, 1, 1, 13, 13, 9, 9, 13, 13

Offset: 0

Marc LeBrun

Another way to construct the array: construct an infinite square matrix starting in the top left corner using the rule that each entry is the smallest nonnegative number that is not in the row to your left or in the column above you.
After a few moves the [symmetric] matrix looks like this:
  0 1 2 3 4 5 ...
  1 0 3 2 5 ...
  2 3 0 1 ?
  3 2 1
  4 5 ?
  5
The ? is then replaced with a 6.

			Table begins
   0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, ...
   1,  0,  3,  2,  5,  4,  7,  6,  9,  8, 11, 10, ...
   2,  3,  0,  1,  6,  7,  4,  5, 10, 11,  8, ...
   3,  2,  1,  0,  7,  6,  5,  4, 11, 10, ...
   4,  5,  6,  7,  0,  1,  2,  3, 12, ...
   5,  4,  7,  6,  1,  0,  3,  2, ...
   6,  7,  4,  5,  2,  3,  0, ...
   7,  6,  5,  4,  3,  2, ...
   8,  9, 10, 11, 12, ...
   9,  8, 11, 10, ...
  10, 11,  8, ...
  11, 10, ...
  12, ...
  ...
The first few antidiagonals are
   0;
   1,  1;
   2,  0,  2;
   3,  3,  3,  3;
   4,  2,  0,  2,  4;
   5,  5,  1,  1,  5,  5;
   6,  4,  6,  0,  6,  4,  6;
   7,  7,  7,  7,  7,  7,  7,  7;
   8,  6,  4,  6,  0,  6,  4,  6,  8;
   9,  9,  5,  5,  1,  1,  5,  5,  9,  9;
  10,  8, 10,  4,  2,  0,  2,  4, 10,  8, 10;
  11, 11, 11, 11,  3,  3,  3,  3, 11, 11, 11, 11;
  12, 10,  8, 10, 12,  2,  0,  2, 12, 10,  8, 10, 12;
  ...
[Symmetric] matrix in base 2:
     0    1   10   11  100  101,  110  111 1000 1001 1010 1011 ...
     1    0   11   10  101  100,  111  110 1001 1000 1011  ...
    10   11    0    1  110  111,  100  101 1010 1011  ...
    11   10    1    0  111  110,  101  100 1011  ...
   100  101  110  111    0    1    10   11  ...
   101  100  111  110    1    0    11  ...
   110  111  100  101   10   11   ...
   111  110  101  100   11  ...
  1000 1001 1010 1011  ...
  1001 1000 1011  ...
  1010 1011  ...
  1011  ...
   ...

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 60.
J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.
Eric Friedman, Scott M. Garrabrant, Ilona K. Phipps-Morgan, A. S. Landsberg and Urban Larsson, Geometric analysis of a generalized Wythoff game, in Games of no Chance 5, MSRI publ. Cambridge University Press, date?
D. Gale, Tracking the Automatic Ant and Other Mathematical Explorations, A Collection of Mathematical Entertainments Columns from The Mathematical Intelligencer, Springer, 1998; see p. 190. [From N. J. A. Sloane, Jul 14 2009]
R. K. Guy, Impartial games, pp. 35-55 of Combinatorial Games, ed. R. K. Guy, Proc. Sympos. Appl. Math., 43, Amer. Math. Soc., 1991.

T. D. Noe, Rows n = 0..100 of triangle, flattened
J.-P. Allouche and J. Shallit, The Ring of k-regular Sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
Rémy Sigrist, Colored representation of T(x,y) for x = 0..1023 and y = 0..1023 (where the hue is function of T(x,y) and black pixels correspond to zeros)
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, The OEIS: A Fingerprint File for Mathematics, arXiv:2105.05111 [math.HO], 2021. Mentions this sequence.
Index entries for sequences related to Nim-sums

Initial rows are A001477, A004442, A004443, A004444, etc. Cf. A051775, A051776.
Cf. A003986 (OR), A004198 (AND), A221146 (carries).
Antidiagonal sums are in A006582.

nimsum := proc(a,b) local t1,t2,t3,t4,l; t1 := convert(a+2^20,base,2); t2 := convert(b+2^20,base,2); t3 := evalm(t1+t2); map(x->x mod 2, t3); t4 := convert(evalm(%),list); l := convert(t4,base,2,10); sum(l[k]*10^(k-1), k=1..nops(l)); end; # memo: adjust 2^20 to be much bigger than a and b
AT := array(0..N,0..N); for a from 0 to N do for b from a to N do AT[a,b] := nimsum(a,b); AT[b,a] := AT[a,b]; od: od:
# alternative:
read("transforms") :
A003987 := proc(n,m)
    XORnos(n,m) ;
end proc: # R. J. Mathar, Apr 17 2013
seq(seq(Bits:-Xor(k,m-k),k=0..m),m=0..20); # Robert Israel, Dec 31 2015

Flatten[Table[BitXor[b, a - b], {a, 0, 10}, {b, 0, a}]] (* BitXor and Nim Sum are equivalent *)

tabl(nn) = {for(n=0, nn, for(k=0, n, print1(bitxor(k, n - k),", ");); print(););};
tabl(13) \\ Indranil Ghosh, Mar 31 2017

for n in range(14):
    print([k^(n - k) for k in range(n + 1)]) # Indranil Ghosh, Mar 31 2017

T(2i,2j) = 2T(i,j), T(2i+1,2j) = 2T(i,j) + 1.

A004198 Table of x AND y, where (x,y) = (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),...

Original entry on oeis.org

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

Offset: 0

David W. Wilson

Or, table of AND(i,j), i >= 0, j >= 0, read by antidiagonals. - N. J. A. Sloane, Feb 08 2016

Or, table of (i+j-Nimsum(i,j))/2 read by antidiagonals [Winning Ways, p. 75]. - N. J. A. Sloane, Feb 22 2019

			The AND(i,j) table (shown without commas or spaces) begins:
0000000000000000000000000...
0101010101010101010101010...
0022002200220022002200220...
0123012301230123012301230...
0000444400004444000044440...
0101454501014545010145450...
0022446600224466002244660...
0123456701234567012345670...
0000000088888888000000008...
0101010189898989010101018...
...
The first few antidiagonals are:
0,
0, 0,
0, 1, 0,
0, 0, 0, 0,
0, 1, 2, 1, 0,
0, 0, 2, 2, 0, 0,
0, 1, 0, 3, 0, 1, 0,
0, 0, 0, 0, 0, 0, 0, 0,
0, 1, 2, 1, 4, 1, 2, 1, 0,
0, 0, 2, 2, 4, 4, 2, 2, 0, 0,
0, 1, 0, 3, 4, 5, 4, 3, 0, 1, 0,
...
- _N. J. A. Sloane_, Feb 08 2016

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 75.

T. D. Noe, Rows n=0..100 of triangle, flattened

Cf. A003986 (OR) and A003987 (XOR). Cf. also A075173, A075175, A221146.

#include 
int main()
{
int n, k;
for (n=0; n<=20; n++){
    for(k=0; k<=n; k++){
        printf("%d, ", (k&(n - k)));
    }
    printf("\n");
}
return 0;
} /* Indranil Ghosh, Apr 01 2017 */

# Maple code for first M rows and columns of AND(i,j) table
M:=24;
f1:=n->[seq(ANDnos(i,n),i=0..M-1)];
for n from 0 to M-1 do lprint(f1(n)); od:
# N. J. A. Sloane, Feb 08 2016

Table[BitAnd[k, n - k], {n, 0, 20}, {k, 0, n}] // Flatten (* Indranil Ghosh, Apr 01 2017 *)

tabl(nn) = {for(n=0, nn, for(k=0, n, print1(bitand(k, n - k), ", "); ); print(); ); };
tabl(20) \\ Indranil Ghosh, Apr 01 2017

for n in range(21):
    print([k&(n - k) for k in range(n + 1)])
# Indranil Ghosh, Apr 01 2017

A087207 A binary representation of the primes that divide a number, shown in decimal.

Original entry on oeis.org

0, 1, 2, 1, 4, 3, 8, 1, 2, 5, 16, 3, 32, 9, 6, 1, 64, 3, 128, 5, 10, 17, 256, 3, 4, 33, 2, 9, 512, 7, 1024, 1, 18, 65, 12, 3, 2048, 129, 34, 5, 4096, 11, 8192, 17, 6, 257, 16384, 3, 8, 5, 66, 33, 32768, 3, 20, 9, 130, 513, 65536, 7, 131072, 1025, 10, 1, 36, 19, 262144, 65, 258

Offset: 1

Mitch Cervinka (puritan(AT)planetkc.com), Oct 26 2003

The binary representation of a(n) shows which prime numbers divide n, but not the multiplicities. a(2)=1, a(3)=10, a(4)=1, a(5)=100, a(6)=11, a(10)=101, a(30)=111, etc.
For n > 1, a(n) gives the (one-based) index of the column where n is located in array A285321. A008479 gives the other index. - Antti Karttunen, Apr 17 2017
From Antti Karttunen, Jun 18 & 20 2017: (Start)
A268335 gives all n such that a(n) = A248663(n); the squarefree numbers (A005117) are all the n such that a(n) = A285330(n) = A048675(n).
For all n > 1 for which the value of A285331(n) is well-defined, we have A285331(a(n)) <= floor(A285331(n)/2), because then n is included in the binary tree A285332 and a(n) is one of its ancestors (in that tree), and thus must be at least one step nearer to its root than n itself.
Conjecture: Starting at any n and iterating the map n -> a(n), we will always reach 0 (see A288569). This conjecture is equivalent to the conjecture that at any n that is neither a prime nor a power of two, we will eventually hit a prime number (which then becomes a power of two in the next iteration). If this conjecture is false then sequence A285332 cannot be a permutation of natural numbers. On the other hand, if the conjecture is true, then A285332 must be a permutation of natural numbers, because all primes and powers of 2 occur in definite positions in that tree. This conjecture also implies the conjectures made in A019565 and A285320 that essentially claim that there are neither finite nor infinite cycles in A019565.
If there are any 2-cycles in this sequence, then both terms of the cycle should be present in A286611 and the larger one should be present in A286612.
(End)
Binary rank of the distinct prime indices of n, where the binary rank of an integer partition y is given by Sum_i 2^(y_i-1). For all prime indices (with multiplicity) we have A048675. - Gus Wiseman, May 25 2024

			a(38) = 129 because 38 = 2*19 = prime(1)*prime(8) and 129 = 2^0 + 2^7 (in binary 10000001).
a(140) = 13, binary 1101 because 140 is divisible by the first, third and fourth primes and 2^(1-1) + 2^(3-1) + 2^(4-1) = 13.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000 [First 1000 terms from _T. D. Noe_]
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization

For partial sums see A288566.
Cf. A000040, A000120, A001221, A005117, A008479, A019565, A055396, A285320, A285321, A285329, A285330, A285332.
Sequences with related definitions: A007947, A008472, A027748, A048675, A248663, A276379 (same sequence shown in base 2), A288569, A289271, A297404.
Cf. A286608 (numbers n for which a(n) < n), A286609 (n for which a(n) > n), and also A286611, A286612.
A003986, A003961, A059896 are used to express relationship between terms of this sequence.
Related to A267116 via A225546.
Positions of particular values are: A000079\{1} (1), A000244\{1} (2), A033845 (3), A000351\{1} (4), A033846 (5), A033849 (6), A143207 (7), A000420\{1} (8), A033847 (9), A033850 (10), A033851 (12), A147576 (14), A147571 (15), A001020\{1} (16), A033848 (17).
A048675 gives binary rank of prime indices.
A061395 gives greatest prime index, least A055396.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Binary indices (listed A048793):
- length A000120, complement A023416
- min A001511, opposite A000012
- sum A029931, product A096111
- max A029837 or A070939, opposite A070940
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
- opposite A371572, sum A230877
Cf. A000720, A005940, A018819, A023506, A071814, A225620, A277319, A277905, A304818, A372689, A372890.

a087207 = sum . map ((2 ^) . (subtract 1) . a049084) . a027748_row
-- Reinhard Zumkeller, Jul 16 2013

a[n_] := Total[ 2^(PrimePi /@ FactorInteger[n][[All, 1]] - 1)]; a[1] = 0; Table[a[n], {n, 1, 69}] (* Jean-François Alcover, Dec 12 2011 *)

a(n) = {if (n==1, 0, my(f=factor(n), v = []); forprime(p=2, vecmax(f[,1]), v = concat(v, vecsearch(f[,1], p)!=0);); fromdigits(Vecrev(v), 2));} \\ Michel Marcus, Jun 05 2017

A087207(n)=vecsum(apply(p->1<M. F. Hasler, Jun 23 2017

from sympy import factorint, primepi
def a(n):
    return sum(2**primepi(i - 1) for i in factorint(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 06 2017

(definec (A087207 n) (if (= 1 n) 0 (+ (A000079 (+ -1 (A055396 n))) (A087207 (A028234 n))))) ;; This uses memoization-macro definec
(define (A087207 n) (A048675 (A007947 n))) ;; Needs code from A007947 and A048675. - Antti Karttunen, Jun 19 2017

Additive with a(p^e) = 2^(i-1) where p is the i-th prime. - Vladeta Jovovic, Oct 29 2003
a(n) gives the m such that A019565(m) = A007947(n). - Naohiro Nomoto, Oct 30 2003
A000120(a(n)) = A001221(n); a(n) = Sum(2^(A049084(p)-1): p prime-factor of n). - Reinhard Zumkeller, Nov 30 2003
G.f.: Sum_{k>=1} 2^(k-1)*x^prime(k)/(1-x^prime(k)). - Franklin T. Adams-Watters, Sep 01 2009
From Antti Karttunen, Apr 17 2017, Jun 19 2017 & Dec 06 2018: (Start)
a(n) = A048675(A007947(n)).
a(1) = 0; for n > 1, a(n) = 2^(A055396(n)-1) + a(A028234(n)).
A000035(a(n)) = 1 - A000035(n). [a(n) and n are of opposite parity.]
A248663(n) <= a(n) <= A048675(n). [XOR-, OR- and +-variants.]
a(A293214(n)) = A218403(n).
a(A293442(n)) = A267116(n).
A069010(a(n)) = A287170(n).
A007088(a(n)) = A276379(n).
A038374(a(n)) = A300820(n) for n >= 1.
(End)
From Peter Munn, Jan 08 2020: (Start)
a(A059896(n,k)) = a(n) OR a(k) = A003986(a(n), a(k)).
a(A003961(n)) = 2*a(n).
a(n^2) = a(n).
a(n) = A267116(A225546(n)).
a(A225546(n)) = A267116(n).
(End)

More terms from Don Reble, Ray Chandler and Naohiro Nomoto, Oct 28 2003

Name clarified by Antti Karttunen, Jun 18 2017

A269160 Formula for Wolfram's Rule 30 cellular automaton: a(n) = n XOR (2n OR 4n).

Original entry on oeis.org

0, 7, 14, 13, 28, 27, 26, 25, 56, 63, 54, 53, 52, 51, 50, 49, 112, 119, 126, 125, 108, 107, 106, 105, 104, 111, 102, 101, 100, 99, 98, 97, 224, 231, 238, 237, 252, 251, 250, 249, 216, 223, 214, 213, 212, 211, 210, 209, 208, 215, 222, 221, 204, 203, 202, 201, 200, 207, 198, 197, 196, 195, 194, 193, 448, 455, 462

Offset: 0

Antti Karttunen, Feb 20 2016

nonn

Take n, write it in binary, see what Rule 30 would do to that state, convert it to decimal: that is a(n). For example, we can see in A110240 that 7 = 111_2 becomes 25 = 11001_2 under Rule 30, which is shown here by a(7) = 25. - N. J. A. Sloane, Nov 25 2016
The sequence is injective: no value occurs more than once.
Fibbinary numbers (A003714) give all integers n>=0 for which a(n) = A048727(n) and for which a(n) = A269161(n).

Antti Karttunen, Table of n, a(n) for n = 0..16383
Eric Weisstein's World of Mathematics, Rule 30
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A003714, A003986, A003987, A057889, A070939.
Cf. A110240 (iterates starting from 1).
Cf. A269162 (left inverse).
Cf. A269163 (same sequence sorted into ascending order).
Cf. A269164 (values missing from this sequence).
Cf. also A048727, A269161.

a[n_] := BitXor[n, BitOr[2n, 4n]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 23 2016 *)

a(n) = bitxor(n, bitor(2*n, 4*n)); \\ Michel Marcus, Feb 23 2016

def A269160(n): return n^(n<<1 | n<<2) # Chai Wah Wu, Jun 29 2022

(define (A269160 n) (A003987bi n (A003986bi (* 4 n) (* 2 n)))) ;; Where A003986bi and A003987bi are implementation of dyadic functions giving bitwise-OR (A003986) and bitwise-XOR (A003987) of their arguments.

a(n) = n XOR (2n OR 4n) = A003987(n, A003986(2*n, 4*n)).
Other identities. For all n >= 0:
a(2*n) = 2*a(n).
a(n) = A057889(A269161(A057889(n))). [Rule 30 is the mirror image of rule 86.]
A269162(a(n)) = n.
For all n >= 1:
A070939(a(n)) - A070939(n) = 2. [The binary length of a(n) is two bits longer than that of n for all nonzero values.]
G.f.: (3*x + 2*x^2 +x^3)/(1 - x^4) + Sum_{k>=1}(2^(k + 1)*x^(2^(k - 1))/((1 + x^(2^(k + 1)))*(1 - x))). - Miles Wilson, Jan 24 2025

A267116 Bitwise-OR of the exponents of primes in the prime factorization of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 03 2016

nonn

			For n = 4 = 2^2, bitwise-OR of 2 alone is 2, thus a(4) = 2.
For n = 6 = 2^1 * 3^1, when we take a bitwise-or of 1 and 1, we get 1, thus a(6) = 1.
For n = 24 = 2^3 * 3^1, bitwise-or of 3 and 1 ("11" and "01" in binary) gives "11", thus a(24) = 3.
For n = 210 = 2^1 * 3^1 * 5^1 * 7^1, bitwise-or of 1, 1, 1 and 1 gives 1, thus a(210) = 1.
For n = 720 = 2^4 * 3^2 * 5^1, bitwise-or of 4, 2 and 1 ("100", "10" and "1" in binary) gives 7 ("111" in binary), thus a(720) = 7.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n

Cf. A001222, A003986, A007814, A028234, A067029.
Cf. A000290 (indices of even numbers).
Cf. A000037 (indices of odd numbers).
Nonunit terms of A005117, A062503, A113849 give the positions of ones, twos, fours respectively in this sequence.
Sequences with similar definitions: A260728, A267113, A267115 (bitwise-AND) and A268387 (bitwise-XOR of exponents).
Sequences with related analysis: A267114, A268374, A268375, A268376.
Sequences A088529, A136565 and A181591 coincide with a(n) for n: 2 <= n < 24.
A003961, A059896 are used to express relationship between terms of this sequence.
Related to A087207 via A225546.

read("transforms"):
A267116 := proc(n)
    local a,e ;
    a := 0 ;
    for e in ifactors(n)[2] do
        a := ORnos(a,op(2,e)) ;
    end do:
    a ;
end proc: # R. J. Mathar, Feb 16 2021

{0}~Join~Rest@ Array[BitOr @@ Map[Last, FactorInteger@ #] &, 120] (* Michael De Vlieger, Feb 04 2016 *)

a(n)=my(f = factor(n)); my(b = 0); for (k=1, #f~, b = bitor(b, f[k,2]);); b; \\ Michel Marcus, Feb 05 2016

a(n)=if(n>1, fold(bitor, factor(n)[,2]), 0) \\ Charles R Greathouse IV, Aug 04 2016

from functools import reduce
from operator import or_
from sympy import factorint
def A267116(n): return reduce(or_,factorint(n).values(),0) # Chai Wah Wu, Aug 31 2022

a(1) = 0; for n > 1: a(n) = A067029(n) OR a(A028234(n)). [Here OR stands for bitwise-or, A003986.]
Other identities and observations. For all n >= 1:
a(n) = A007814(n) OR A260728(n) OR A267113(n).
a(n) = A001222(n) - A268374(n).
A268387(n) <= a(n) <= A001222(n).
From Peter Munn, Jan 08 2020: (Start)
a(A059896(n,k)) = a(n) OR a(k).
a(A003961(n)) = a(n).
a(n^2) = 2*a(n).
a(n) = A087207(A225546(n)).
a(A225546(n)) = A087207(n).
(End)

A003990 Table of lcm(x,y), read along antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 6, 6, 4, 5, 4, 3, 4, 5, 6, 10, 12, 12, 10, 6, 7, 6, 15, 4, 15, 6, 7, 8, 14, 6, 20, 20, 6, 14, 8, 9, 8, 21, 12, 5, 12, 21, 8, 9, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 11, 10, 9, 8, 35, 6, 35, 8, 9, 10, 11, 12, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 12, 13, 12, 33, 20, 45, 24

Offset: 1

Marc LeBrun

A(x,x) = x on the diagonal. - Reinhard Zumkeller, Aug 05 2012

			The symmetric array is lcm(x,y) = lcm(y,x):
   1  2  3  4  5  6  7  8  9 10 ...
   2  2  6  4 10  6 14  8 18 10 ...
   3  6  3 12 15  6 21 24  9 30 ...
   4  4 12  4 20 12 28  8 36 20 ...
   5 10 15 20  5 30 35 40 45 10 ...
   6  6  6 12 30  6 42 24 18 30 ...
   7 14 21 28 35 42  7 56 63 70 ...
   8  8 24  8 40 24 56  8 72 40 ...
   9 18  9 36 45 18 63 72  9 90 ...
  10 10 30 20 10 30 70 40 90 10 ...

T. D. Noe, First 100 antidiagonals of array, flattened
Kival Ngaokrajang, Illustration of pattern, where terms with least significant decimal digit equal to zero are colored.
Index entries for sequences related to lcm's

Cf. A003989, A003991, A051173.
A(x, y) = A075174(A003986(A075173(x), A075173(y))) = A075176(A003986(A075175(x), A075175(y))).
Antidiagonal sums are in A006580.
Cf. A002260.

a003990 x y = a003990_adiag x !! (y-1)
a003990_adiag n = a003990_tabl !! (n-1)
a003990_tabl = zipWith (zipWith lcm) a002260_tabl $ map reverse a002260_tabl
-- Reinhard Zumkeller, Aug 05 2012

Table[ LCM[x-y, y], {x, 1, 14}, {y, 1, x-1}] // Flatten (* Jean-François Alcover, Aug 20 2013 *)

A(x,y)=lcm(x,y) \\ Charles R Greathouse IV, Feb 06 2017

A057300 Binary counter with odd/even bit positions swapped; base-4 counter with 1's replaced by 2's and vice versa.

Original entry on oeis.org

0, 2, 1, 3, 8, 10, 9, 11, 4, 6, 5, 7, 12, 14, 13, 15, 32, 34, 33, 35, 40, 42, 41, 43, 36, 38, 37, 39, 44, 46, 45, 47, 16, 18, 17, 19, 24, 26, 25, 27, 20, 22, 21, 23, 28, 30, 29, 31, 48, 50, 49, 51, 56, 58, 57, 59, 52, 54, 53, 55, 60, 62, 61, 63, 128, 130, 129, 131, 136, 138

Offset: 0

Marc LeBrun, Aug 24 2000

A self-inverse permutation of the integers.
a(n) = n if and only if n can be written as 3*Sum_{k>=0} d_i*4^k, where d_i is either 0 or 1. - Jon Perry, Oct 06 2012
From Veselin Jungic, Mar 03 2015: (Start)
In 1988 A. F. Sidorenko, see the Sidorenko reference, used this sequence as an example of a permutation of the set of positive integers with the property that if positive integers i, j, and k form a 3-term arithmetic progression then the corresponding terms a(i), a(j), and a(k) do not form an arithmetic progression.
In the terminology introduced in the Brown, Jungic, and Poelstra reference, the sequence does not contain "double 3-term arithmetic progressions".
It is not difficult to check that this sequence is with unbounded gaps, i.e., for any positive number m there is a natural number n such that a(n+1) - a(n) > m.
It is an open question if every sequence of integers with bounded gaps must contain a double 3-term arithmetic progression. This problem is equivalent to the well known additive square problem in infinite words: Is it true that any infinite word with a finite set of integers as its alphabet contains two consecutive blocks of the same length and the same sum? For more details about the additive square problem in infinite words see the following references: Ardal, et al.; Brown and Freedman; Freedman; Grytczuk; Halbeisen and Hungerbuhler, and Pirillo and Varricchio.
The sequence was attributed to Sidorenko in P. Hegarty's paper "Permutations avoiding arithmetic patterns". In his paper Hegarty characterized the countably infinite abelian groups for which there exists a bijection mapping arithmetic progressions to non-arithmetic progressions. This was further generalized by Jungic and Sahasrabudhe. (End)

			a(31) = a(4*7+3) = 4*a(7) + a(3) = 4*11 + 3 = 47.

Paul Tek, Table of n, a(n) for n = 0..16383
H. Ardal, T. Brown, V. Jungic, and J. Sahasrabudhe, On Additive and Abelian Complexity in Infinite Words, INTEGERS: Elect. J. Combin. Number Theory, 12 (2012), A21.
T. C. Brown and A. R. Freedman, Arithmetic progressions in lacunary sets, Rocky Mountain J. Math., 17 Number 3 (1987), 587-596. doi:10.1216/RMJ-1987-17-3-587
T. Brown, V. Jungic, and A. Poelstra, On 3-term double arithmetic progressions, INTEGERS: Elect. J. Combin. Number Theory, 14 (2014), A43.
A. R. Freedman, Sequences on sets of four numbers, INTEGERS: Elect. J. Combin. Number Theory, 16 (2016), A33.
J. Grytczuk, Thue type problems for graphs, points and numbers, Discrete Math., 308 (2008), 4419-4429.
L. Halbeisen and N. Hungerbuhler, An application of van der Waerden's theorem in additive number theory, INTEGERS: Elect. J. Combin. Number Theory, 0 (2000), A7.
Peter Hegarty, Permutations avoiding arithmetic patterns, The Electronic Journal of Combinatorics, 11 (2004), #R39.
V. Jungic and J. Sahasrabudhe, Permutations destroying arithmetic structure, The Electronic Journal of Combinatorics, Volume 22, Issue 2 (2015), Paper #P2.5.
G. Pirillo and S. Varricchio, On uniformly repetitive semigroups, Semigroup Forum, 49 (1994), 125-129.
A. F. Sidorenko, An infinite permutation without arithmetic progressions, Discrete Math., 69 (1988), 211.
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Index entries for sequences related to binary expansion of n
Index entries for sequences that are permutations of the natural numbers

Sequences used in definitions of this sequence: A000695, A059905, A059906.
Sequences with similar definitions: A057301, A126006, A126007, A126008, A163241, A163327.
A003986, A003987, A004198, A053985, A054240 are used to express relationships between sequence terms.

#include 
uint32_t a(uint32_t n) { return ((n & 0x55555555) << 1) | ((n & 0xaaaaaaaa) >> 1); } /* Falk Hüffner, Jan 23 2022 */

a:= proc(n) option remember; `if`(n=0, 0,
      a(iquo(n, 4, 'r'))*4+[0, 2, 1, 3][r+1])
    end:
seq(a(n), n=0..69);  # Alois P. Heinz, Jan 25 2022

Table[FromDigits[IntegerDigits[n,4]/.{1->2,2->1},4],{n,0,70}] (* Harvey P. Dale, Aug 24 2017 *)

A057300(n) = { my(t=1,s=0); while(n>0, if(1==(n%4),n++,if(2==(n%4),n--)); s += (n%4)*t; n >>= 2; t <<= 2); (s); }; \\ Antti Karttunen, Apr 14 2018

Conjecture: a(2*n) = -2*a(n) + 5*n, a(2*n+1) = -2*a(n) + 5*n + 2. - Ralf Stephan, Oct 11 2003
a(4n+k) = 4a(n) + a(k), 0 <= k <= 3. - Jon Perry, Oct 06 2012
a(n) = A000695(A059906(n)) + 2*A000695(A059905(n)). - Antti Karttunen, Apr 14 2018
From Peter Munn, Dec 10 2019: (Start)
a(a(n)) = n.
a(A000695(m) + 2*A000695(n)) = 2*A000695(m) + A000695(n).
a(n OR k) = a(n) OR a(k), where OR is bitwise-or (A003986).
a(n XOR k) = a(n) XOR a(k), where XOR is bitwise exclusive-or (A003987).
a(n AND k) = a(n) AND a(k), where AND is bitwise-and (A004198).
a(A054240(n,k)) = A054240(a(n), a(k)). (End)
a(n) = 5*n/4 - 3*A053985(2*n)/8. - Alan Michael Gómez Calderón, May 20 2025

cp's OEIS Frontend

A374799 Irregular table T(n, k), n >= 0, k = 1..A048883(n), read by rows; the n-th row lists the numbers m such that A003986(m) = n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A269170 a(n) = n OR floor(n/2), where OR is bitwise-OR (A003986).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Scheme

Formula

A007814 Exponent of highest power of 2 dividing n, a.k.a. the binary carry sequence, the ruler sequence, or the 2-adic valuation of n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Haskell

Magma

Maple

Mathematica

PARI

Python

Python

R

Scheme

Formula

Extensions

A003987 Table of n XOR m (or Nim-sum of n and m) read by antidiagonals with m>=0, n>=0.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A004198 Table of x AND y, where (x,y) = (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),...

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

C

Maple

Mathematica

PARI

Python

A087207 A binary representation of the primes that divide a number, shown in decimal.

Views

Author