A181988 -id:A181988 - cp's OEIS frontend

A001511 The ruler function: exponent of the highest power of 2 dividing 2n. Equivalently, the 2-adic valuation of 2n.

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

Offset: 1

N. J. A. Sloane

Number of 2's dividing 2*n.
a(n) is equivalently the exponent of the smallest power of 2 which does not divide n. - David James Sycamore, Oct 02 2023
a(n) - 1 is the number of trailing zeros in the binary expansion of n.
If you are counting in binary and the least significant bit is numbered 1, the next bit is 2, etc., a(n) is the bit that is incremented when increasing from n-1 to n. - Jud McCranie, Apr 26 2004
Number of steps to reach an integer starting with (n+1)/2 and using the map x -> x*ceiling(x) (cf. A073524).
a(n) is the number of the disk to be moved at the n-th step of the optimal solution to Towers of Hanoi problem (comment from Andreas M. Hinz).
Shows which bit to flip when creating the binary reflected Gray code (bits are numbered from the right, offset is 1). This is essentially equivalent to Hinz's comment. - Adam Kertesz, Jul 28 2001
a(n) is the Hamming distance between n and n-1 (in binary). This is equivalent to Kertesz's comments above. - Tak-Shing Chan (chan12(AT)alumni.usc.edu), Feb 25 2003
Let S(0) = {1}, S(n) = {S(n-1), S(n-1)-{x}, x+1} where x = last term of S(n-1); sequence gives S(infinity). - Benoit Cloitre, Jun 14 2003
The sum of all terms up to and including the first occurrence of m is 2^m-1. - Donald Sampson (marsquo(AT)hotmail.com), Dec 01 2003
m appears every 2^m terms starting with the 2^(m-1)th term. - Donald Sampson (marsquo(AT)hotmail.com), Dec 08 2003
Sequence read mod 4 gives A092412. - Philippe Deléham, Mar 28 2004
If q = 2n/2^A001511(n) and if b(m) is defined by b(0)=q-1 and b(m)=2*b(m-1)+1, then 2n = b(A001511(n)) + 1. - Gerald McGarvey, Dec 18 2004
Repeating pattern ABACABADABACABAE ... - Jeremy Gardiner, Jan 16 2005
Relation to C(n) = Collatz function iteration using only odd steps: a(n) is the number of right bits set in binary representation of A004767(n) (numbers of the form 4*m+3). So for m=A004767(n) it follows that there are exactly a(n) recursive steps where mBetween every two instances of any positive integer m there are exactly m distinct values (1 through m-1 and one value greater than m). - Franklin T. Adams-Watters, Sep 18 2006
Number of divisors of n of the form 2^k. - Giovanni Teofilatto, Jul 25 2007
Every prefix up to (but not including) the first occurrence of some k >= 2 is a palindrome. - Gary W. Adamson, Sep 24 2008
1 interleaved with (2 interleaved with (3 interleaved with ( ... ))). - Eric D. Burgess (ericdb(AT)gmail.com), Oct 17 2009
A054525 (Möbius transform) * A001511 = A036987 = A047999^(-1) * A001511. - Gary W. Adamson, Oct 26 2009
Equals A051731 * A036987, (inverse Möbius transform of the Fredholm-Rueppel sequence) = A047999 * A036987. - Gary W. Adamson, Oct 26 2009
Cf. A173238, showing links between generalized ruler functions and A000041. - Gary W. Adamson, Feb 14 2010
Given A000041, P(x) = A(x)/A(x^2) with P(x) = (1 + x + 2x^2 + 3x^3 + 5x^4 + 7x^5 + ...), A(x) = (1 + x + 3x^2 + 4x^3 + 10x^4 + 13x^5 + ...), A(x^2) = (1 + x^2 + 3x^4 + 4x^6 + 10x^8 + ...), where A092119 = (1, 1, 3, 4, 10, ...) = Euler transform of the ruler sequence, A001511. - Gary W. Adamson, Feb 11 2010
Subtracting 1 from every term and deleting any 0's yields the same sequence, A001511. - Ben Branman, Dec 28 2011
In the listing of the compositions of n as lists in lexicographic order, a(k) is the last part of composition(k) for all k <= 2^(n-1) and all n, see example. - Joerg Arndt, Nov 12 2012
According to Hinz, et al. (see links), this sequence was studied by Louis Gros in his 1872 pamphlet "Théorie du Baguenodier" and has therefore been called the Gros sequence.
First n terms comprise least squarefree word of length n using positive integers, where "squarefree" means that the word contains no consecutive identical subwords; e.g., 1 contains no square; 11 contains a square but 12 does not; 121 contains no square; both 1211 and 1212 have squares but 1213 does not; etc. - Clark Kimberling, Sep 05 2013
Length of 0-run starting from 2 (10, 100, 110, 1000, 1010, ...), or length of 1-run starting from 1 (1, 11, 101, 111, 1001, 1011, ...) of every second number, from right to left in binary representation. - Armands Strazds, Apr 13 2017
a(n) is also the frequency of the largest part in the integer partition having viabin number n. The viabin number of an integer partition is defined in the following way. Consider the southeast border of the Ferrers board of the integer partition and consider the binary number obtained by replacing each east step with 1 and each north step, except the last one, with 0. The corresponding decimal form is, by definition, the viabin number of the given integer partition. "Viabin" is coined from "via binary". For example, consider the integer partition [2,2,2,1]. The southeast border of its Ferrers board yields 10100, leading to the viabin number 20. - Emeric Deutsch, Jul 24 2017
As A000005(n) equals the number of even divisors of 2n and A001227(n) = A001227(2n), the formula A001511(n) = A000005(n)/A001227(n) might be read as "The number of even divisors of 2n is always divisible by the number of odd divisors of 2n" (where number of divisors means sum of zeroth powers of divisors). Conjecture: For any nonnegative integer k, the sum of the k-th powers of even divisors of n is always divisible by the sum of the k-th powers of odd divisors of n. - Ivan N. Ianakiev, Jul 06 2019
From Benoit Cloitre, Jul 14 2022: (Start)
To construct the sequence, start from 1's separated by a place 1,,1,,1,,1,,1,,1,,1,,1,,1,,1,,1,,1,,1,,1,...
Then put the 2's in every other remaining place
  1,2,1,,1,2,1,,1,2,1,,1,2,1,,1,2,1,,1,2,1,,1,2,1,...
Then the 3's in every other remaining place
  1,2,1,3,1,2,1,,1,2,1,3,1,2,1,,1,2,1,3,1,2,1,,1,2,1,...
Then the 4's in every other remaining place
  1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,,1,2,1,3,1,2,1,4,1,2,1,...
By iterating this process, we get the ruler function 1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,... (End)
a(n) is the least positive 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
a(n) is the smallest positive integer that does not occur in the coincidences of the sequence so far a(1..n-1) and its reverse. - Neal Gersh Tolunsky, Jan 18 2023
The geometric mean of this sequence approaches the Somos constant (A112302). - Jwalin Bhatt, Jan 31 2025

			For example, 2^1|2, 2^2|4, 2^1|6, 2^3|8, 2^1|10, 2^2|12, ... giving the initial terms 1, 2, 1, 3, 1, 2, ...
From _Omar E. Pol_, Jun 12 2009: (Start)
Triangle begins:
1;
2,1;
3,1,2,1;
4,1,2,1,3,1,2,1;
5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1;
6,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1;
7,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,6,1,2,1,3,...
(End)
S(0) = {} S(1) = 1 S(2) = 1, 2, 1 S(3) = 1, 2, 1, 3, 1, 2, 1 S(4) = 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1. - Yann David (yann_david(AT)hotmail.com), Mar 21 2010
From _Joerg Arndt_, Nov 12 2012: (Start)
The 16 compositions of 5 as lists in lexicographic order:
[ n]  a(n)  composition
[ 1]  [ 1]  [ 1 1 1 1 1 ]
[ 2]  [ 2]  [ 1 1 1 2 ]
[ 3]  [ 1]  [ 1 1 2 1 ]
[ 4]  [ 3]  [ 1 1 3 ]
[ 5]  [ 1]  [ 1 2 1 1 ]
[ 6]  [ 2]  [ 1 2 2 ]
[ 7]  [ 1]  [ 1 3 1 ]
[ 8]  [ 4]  [ 1 4 ]
[ 9]  [ 1]  [ 2 1 1 1 ]
[10]  [ 2]  [ 2 1 2 ]
[11]  [ 1]  [ 2 2 1 ]
[12]  [ 3]  [ 2 3 ]
[13]  [ 1]  [ 3 1 1 ]
[14]  [ 2]  [ 3 2 ]
[15]  [ 1]  [ 4 1 ]
[16]  [ 5]  [ 5 ]
a(n) is the last part in each list.
(End)
From _Omar E. Pol_, Aug 20 2013: (Start)
Also written as a triangle in which the right border gives A000027 and row lengths give A011782 and row sums give A000079 the sequence begins:
1;
2;
1,3;
1,2,1,4;
1,2,1,3,1,2,1,5;
1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,6;
1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,7;
(End)
G.f. = x + 2*x^2 + x^3 + 3*x^4 + x^5 + 2*x^6 + x^7 + 4*x^8 + x^9 + 2*x^10 + ...

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003.
E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 2nd ed., 2001-2003; see Dim- and Dim+ on p. 98; Dividing Rulers, on pp. 436-437; The Ruler Game, pp. 469-470; Ruler Fours, Fives, ... Fifteens on p. 470.
L. Gros, Théorie du Baguenodier, Aimé Vingtrinier, Lyon, 1872.
R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section E22.
A. M. Hinz, The Tower of Hanoi, in Algebras and combinatorics (Hong Kong, 1997), 277-289, Springer, Singapore, 1999.
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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
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B. Baker Swart, R. Florez, D. Narayan, and G. Rudolph Extrema Property of the k-Ranking of Directed Paths and Cycles, AKCE International Journal of Graphs and Combinatorics, 13 (2016) 38-53.
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Antonin Callard, Léo Paviet Salomon, and Pascal Vanier, Computability of extender sets in multidimensional subshifts, arXiv:2401.07549 [cs.DM], 2024.
Imanuel Chen and Michael Z. Spivey, Integral Generalized Binomial Coefficients of Multiplicative Functions, Preprint 2015; Summer Research Paper 238, Univ. Puget Sound.
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Madeleine Goertz and Aaron Williams, The Quaternary Gray Code and How It Can Be Used to Solve Ziggurat and Other Ziggu Puzzles, arXiv:2411.19291 [math.CO], 2024. See pp. 1, 2, 12, 24.
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Michael Naylor, Abacaba-Dabacaba [updated by _Jeremy Gardiner_, Sep 11 2015]
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Eric Weisstein's World of Mathematics, Binary Carry Sequence, Ruler Function, and Towers of Hanoi
Index entries for "core" sequences
Index entries for sequences related to binary expansion of n
Index entries for sequences that are fixed points of mappings
Index entries for sequences related to Towers of Hanoi

Column 1 of table A050600.
Sequence read mod 2 gives A035263.
Sequence is bisection of A007814, A050603, A050604, A067029, A089309.
This is Guy Steele's sequence GS(4, 2) (see A135416).
Cf. A005187 (partial sums), A085058 (bisection), A112302 (geometric mean), A171977 (2^a(n)).
Cf. A000005, A000041, A000079, A003188, A003278, A003602, A007949, A018238, A047999, A051731, A054525, A054852, A065176, A082850, A089080, A092119, A117303, A129360, A130093, A173238, A181988, A220466.
Cf. A287896, A002487, A209229 (Mobius trans.), A092673 (Dirichlet inv.).
Cf. generalized ruler functions for k=3,4,5: A051064, A115362, A055457.

a001511 n = length $ takeWhile ((== 0) . (mod n)) a000079_list
-- Reinhard Zumkeller, Sep 27 2011

a001511 n | odd n = 1 | otherwise = 1 + a001511 (n `div` 2)
-- Walt Rorie-Baety, Mar 22 2013

nmax=5;r=1;for n=2:nmax;r=[r n r];end % Adriano Caroli, Feb 26 2016

[Valuation(2*n,2): n in [1..105]]; // Bruno Berselli, Nov 23 2015

A001511 := n->2-wt(n)+wt(n-1); # where wt is defined in A000120
# This is the binary logarithm of the denominator of (256^n-1)B_{8n}/n, in Maple parlance a := n -> log[2](denom((256^n-1)*bernoulli(8*n)/n)). - Peter Luschny, May 31 2009
A001511 := n -> padic[ordp](2*n,2): seq(A001511(n), n=1..105);  # Peter Luschny, Nov 26 2010
a:= n-> ilog2((Bits[Xor](2*n, 2*n-1)+1)/2): seq(a(n), n=1..50);  # Gary Detlefs, Dec 13 2018

Array[ If[ Mod[ #, 2] == 0, FactorInteger[ # ][[1, 2]], 0] &, 105] + 1 (* or *)
Nest[ Flatten[ # /. a_Integer -> {1, a + 1}] &, {1}, 7] (* Robert G. Wilson v, Mar 04 2005 *)
IntegerExponent[2*n, 2] (* Alexander R. Povolotsky, Aug 19 2011 *)
myHammingDistance[n_, m_] := Module[{g = Max[m, n], h = Min[m, n]}, b1 = IntegerDigits[g, 2]; b2 = IntegerDigits[h, 2, Length[b1]]; HammingDistance[b1, b2]] (* Vladimir Shevelev A206853 *) Table[ myHammingDistance[n, n - 1], {n, 111}] (* Robert G. Wilson v, Apr 05 2012 *)
Table[Position[Reverse[IntegerDigits[n,2]],1,1,1],{n,110}]//Flatten (* Harvey P. Dale, Aug 18 2017 *)

a(n) = sum(k=0,floor(log(n)/log(2)),floor(n/2^k)-floor((n-1)/2^k)) /* Ralf Stephan */

a(n)=if(n%2,1,factor(n)[1,2]+1) /* Jon Perry, Jun 06 2004 */

{a(n) = if( n, valuation(n, 2) + 1, 0)}; /* Michael Somos, Sep 30 2006 */

{a(n)=if(n==1,1,polcoeff(x-sum(k=1, n-1, a(k)*x^k*(1-x^k)*(1-x+x*O(x^n))), n))} /* Paul D. Hanna, Jun 22 2007 */

def a(n): return bin(n)[2:][::-1].index("1") + 1 # Indranil Ghosh, May 11 2017

A001511 = lambda n: (n&-n).bit_length() # M. F. Hasler, Apr 09 2020

def A001511(n): return (~n & n-1).bit_length()+1 # Chai Wah Wu, Jul 01 2022

[valuation(2*n,2) for n in (1..105)]  # Bruno Berselli, Nov 23 2015

(define (A001511 n) (let loop ((n n) (e 1)) (if (odd? n) e (loop (/ n 2) (+ 1 e))))) ;; Antti Karttunen, Oct 06 2017

a(n) = A007814(n) + 1 = A007814(2*n).
a(2*n+1) = 1; a(2*n) = 1 + a(n). - Philippe Deléham, Dec 08 2003
a(n) = 2 - A000120(n) + A000120(n-1), n >= 1. - Daniele Parisse
a(n) = 1 + log_2(abs(A003188(n) - A003188(n-1))).
Multiplicative with a(p^e) = e+1 if p = 2; 1 if p > 2. - David W. Wilson, Aug 01 2001
For any real x > 1/2: lim_{N->infinity} (1/N)*Sum_{n=1..N} x^(-a(n)) = 1/(2*x-1); also lim_{N->infinity} (1/N)*Sum_{n=1..N} 1/a(n) = log(2). - Benoit Cloitre, Nov 16 2001
s(n) = Sum_{k=1..n} a(k) is asymptotic to 2*n since s(n) = 2*n - A000120(n). - Benoit Cloitre, Aug 31 2002
For any n >= 0, for any m >= 1, a(2^m*n + 2^(m-1)) = m. - Benoit Cloitre, Nov 24 2002
a(n) = Sum_{d divides n and d is odd} mu(d)*tau(n/d). - Vladeta Jovovic, Dec 04 2002
G.f.: A(x) = Sum_{k>=0} x^(2^k)/(1-x^(2^k)). - Ralf Stephan, Dec 24 2002
a(1) = 1; for n > 1, a(n) = a(n-1) + (-1)^n*a(floor(n/2)). - Vladeta Jovovic, Apr 25 2003
A fixed point of the mapping 1->12; 2->13; 3->14; 4->15; 5->16; ... . - Philippe Deléham, Dec 13 2003
Product_{k>0} (1+x^k)^a(k) is g.f. for A000041(). - Vladeta Jovovic, Mar 26 2004
G.f. A(x) satisfies A(x) = A(x^2) + x/(1-x). - Franklin T. Adams-Watters, Feb 09 2006
a(A118413(n,k)) = A002260(n,k); = a(A118416(n,k)) = A002024(n,k); a(A014480(n)) = A003602(A014480(n)). - Reinhard Zumkeller, Apr 27 2006
Ordinal transform of A003602. - Franklin T. Adams-Watters, Aug 28 2006 (The ordinal transform of a sequence b_0, b_1, b_2, ... is the sequence a_0, a_1, a_2, ... where a_n is the number of times b_n has occurred in {b_0 ... b_n}.)
Could be extended to n <= 0 using a(-n) = a(n), a(0) = 0, a(2*n) = a(n)+1 unless n=0. - Michael Somos, Sep 30 2006
A094267(2*n) = A050603(2*n) = A050603(2*n + 1) = a(n). - Michael Somos, Sep 30 2006
Sequence = A129360 * A000005 = M*V, where M = an infinite lower triangular matrix and V = d(n) as a vector: [1, 2, 2, 3, 2, 4, ...]. - Gary W. Adamson, Apr 15 2007
Row sums of triangle A130093. - Gary W. Adamson, May 13 2007
Dirichlet g.f.: zeta(s)*2^s/(2^s-1). - Ralf Stephan, Jun 17 2007
a(n) = -Sum_{d divides n} mu(2*d)*tau(n/d). - Benoit Cloitre, Jun 21 2007
G.f.: x/(1-x) = Sum_{n>=1} a(n)*x^n*( 1 - x^n ). - Paul D. Hanna, Jun 22 2007
2*n = 2^a(n)* A000265(n). - Eric Desbiaux, May 14 2009 [corrected by Alejandro Erickson, Apr 17 2012]
Multiplicative with a(2^k) = k + 1, a(p^k) = 1 for any odd prime p. - Franklin T. Adams-Watters, Jun 09 2009
With S(n): 2^n - 1 first elements of the sequence then S(0) = {} (empty list) and if n > 0, S(n) = S(n-1), n, S(n-1). - Yann David (yann_david(AT)hotmail.com), Mar 21 2010
a(n) = log_2(A046161(n)/A046161(n-1)). - Johannes W. Meijer, Nov 04 2012
a((2*n-1)*2^p) = p+1, p >= 0 and n >= 1. - Johannes W. Meijer, Feb 05 2013
a(n+1) = 1 + Sum_{j=0..ceiling(log_2(n+1))} (j * (1 - abs(sign((n mod 2^(j + 1)) - 2^j + 1)))). - Enrico Borba, Oct 01 2015
Conjecture: a(n) = A181988(n)/A003602(n). - L. Edson Jeffery, Nov 21 2015
a(n) = log_2(A006519(n)) + 1. - Doug Bell, Jun 02 2017
Inverse Moebius transform of A209229. - Andrew Howroyd, Aug 04 2018
a(n) = 1 + (A183063(n)/A001227(n)). - Omar E. Pol, Nov 06 2018 (after Franklin T. Adams-Watters)
a(n) = log_2((Xor(2*n,2*n-1)+1)/2). - Gary Detlefs, Dec 13 2018
(2^(a(n)-1)-1)*(n mod 4) = 2*floor(((n+1) mod 4)/3). - Gary Detlefs, Dec 14 2018
a(n) = A000005(n)/A001227(n). - Ivan N. Ianakiev, Jul 05 2019
a(n) = Sum_{j=1..r} (j/2^j)*(Product_{k=1..j} (1 - (-1)^floor( (n+2^(j-1))/2^(k-1) ))), for n < a predefined 2^r. - Adriano Caroli, Sep 30 2019

Name edited following suggestion by David James Sycamore, Oct 05 2023

A003602 Kimberling's paraphrases: if n = (2k-1)*2^m then a(n) = k.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8, 1, 9, 5, 10, 3, 11, 6, 12, 2, 13, 7, 14, 4, 15, 8, 16, 1, 17, 9, 18, 5, 19, 10, 20, 3, 21, 11, 22, 6, 23, 12, 24, 2, 25, 13, 26, 7, 27, 14, 28, 4, 29, 15, 30, 8, 31, 16, 32, 1, 33, 17, 34, 9, 35, 18, 36, 5, 37, 19, 38, 10, 39, 20, 40, 3, 41, 21, 42

Offset: 1

N. J. A. Sloane, Mira Bernstein

Fractal sequence obtained from powers of 2.
k occurs at (2*k-1)*A000079(m), m >= 0. - Robert G. Wilson v, May 23 2006
Sequence is T^(oo)(1) where T is acting on a word w = w(1)w(2)..w(m) as follows: T(w) = "1"w(1)"2"w(2)"3"(...)"m"w(m)"m+1". For instance T(ab) = 1a2b3. Thus T(1) = 112, T(T(1)) = 1121324, T(T(T(1))) = 112132415362748. - Benoit Cloitre, Mar 02 2009
Note that iterating the post-numbering operator U(w) = w(1) 1 w(2) 2 w(3) 3... produces the same limit sequence except with an additional "1" prepended, i.e., 1,1,1,2,1,3,2,4,... - Glen Whitney, Aug 30 2023
In the binary expansion of n, first swallow all zeros from the right, then add 1, and swallow the now-appearing 0 bit as well. - Ralf Stephan, Aug 22 2013
Although A264646 and this sequence initially agree in their digit-streams, they differ after 48 digits. - N. J. A. Sloane, Nov 20 2015
"[This is a] fractal because we get the same sequence after we delete from it the first appearance of all positive integers" - see Cobeli and Zaharescu link. - Robert G. Wilson v, Jun 03 2018
From Peter Munn, Jun 16 2022: (Start)
The sequence is the list of positive integers interleaved with the sequence itself. Provided the offset is suitable (which is the case here) a term of such a self-interleaved sequence is determined by the odd part of its index. Putting some of the formulas given here into words, a(n) is the position of the odd part of n in the list of odd numbers.
Applying the interleaving transform again, we get A110963.
(End)
Omitting all 1's leaves A131987 + 1. - David James Sycamore, Jul 26 2022
a(n) is also the smallest positive number not among the terms between a(a(n-1)) and a(n-1) inclusive (with a(0)=1 prepended). - Neal Gersh Tolunsky, Mar 07 2023

			From _Peter Munn_, Jun 14 2022: (Start)
Start of table showing the interleaving with the positive integers:
   n  a(n)  (n+1)/2  a(n/2)
   1    1      1
   2    1               1
   3    2      2
   4    1               1
   5    3      3
   6    2               2
   7    4      4
   8    1               1
   9    5      5
  10    3               3
  11    6      6
  12    2               2
(End)

Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
Cristian Cobeli and Alexandru Zaharescu, Promenade around Pascal Triangle - Number Motives, Bull. Math. Soc. Sci. Math. Roumanie, Tome 56(104) No. 1, 2013, 73-98.
J.-P. Delahaye, La marelle arithmétique, Pour la Science, No. 360, October 2007. In French.
Dale Gerdemann, Plotting Adjacent Points in A003602, Kimberling's Paraphrase, YouTube Video, 2015.
Dale Gerdemann, Plotting Adjacent Terms of A003602 Modulo Increasing Powers of 2, YouTube Video, 2015.
Douglas E. Iannucci and Urban Larsson, Game values of arithmetic functions, arXiv:2101.07608 [math.NT], 2021.
Jonas Kaiser, On the relationship between the Collatz conjecture and Mersenne prime numbers, arXiv preprint arXiv:1608.00862 [math.GM], 2016.
Clark Kimberling, Numeration systems and fractal sequences, Acta Arithmetica 73 (1995) 103-117.
Clark Kimberling, Fractal sequences
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Matty van-Son, Palindromic sequences of the Markov spectrum, arXiv:1804.10802 [math.NT], 2018.
Eric Weisstein's World of Mathematics, Odd Part
Index entries for sequences related to binary expansion of n

a(n) is the index of the column in A135764 where n appears (see also A054582).
Cf. A000079, A000265, A001511, A003603, A003961, A014577 (with offset 1, reduction mod 2), A025480, A035528, A048673, A101279, A110963, A117303, A126760, A181988, A220466, A249745, A253887, A337821 (2-adic valuation).
Cf. also A349134 (Dirichlet inverse), A349135 (sum with it), A349136 (Möbius transform), A349431, A349371 (inverse Möbius transform).
Cf. A065120, A131987.
Cf. A264646.
Cf. A123390, A118319.

a003602 = (`div` 2) . (+ 1) . a000265
-- Reinhard Zumkeller, Feb 16 2012, Oct 14 2010

import Data.List (transpose)
a003602 = flip div 2 . (+ 1) . a000265
a003602_list = concat $ transpose [[1..], a003602_list]
-- Reinhard Zumkeller, Aug 09 2013, May 23 2013

A003602:=proc(n) options remember: if n mod 2 = 1 then RETURN((n+1)/2) else RETURN(procname(n/2)) fi: end proc:
seq(A003602(n), n=1..83); # Pab Ter
nmax := 83: for m from 0 to ceil(simplify(log[2](nmax))) do for k from 1 to ceil(nmax/(m+2)) do a((2*k-1)*2^m) := k od: od: seq(a(k), k=1..nmax); # Johannes W. Meijer, Feb 04 2013
A003602 := proc(n)
    a := 1;
    for p in ifactors(n)[2] do
        if op(1,p) > 2 then
            a := a*op(1,p)^op(2,p) ;
        end if;
    end do  :
    (a+1)/2 ;
end proc: # R. J. Mathar, May 19 2016

a[n_] := Block[{m = n}, While[ EvenQ@m, m /= 2]; (m + 1)/2]; Array[a, 84] (* or *)
a[1] = 1; a[n_] := a[n] = If[OddQ@n, (n + 1)/2, a[n/2]]; Array[a, 84] (* Robert G. Wilson v, May 23 2006 *)
a[n_] := Ceiling[NestWhile[Floor[#/2] &, n, EvenQ]/2]; Array[a, 84] (* Birkas Gyorgy, Apr 05 2011 *)
a003602 = {1}; max = 7; Do[b = {}; Do[AppendTo[b, {k, a003602[[k]]}], {k, Length[a003602]}]; a003602 = Flatten[b], {n, 2, max}]; a003602 (* L. Edson Jeffery, Nov 21 2015 *)

A003602(n)=(n/2^valuation(n,2)+1)/2; /* Joerg Arndt, Apr 06 2011 */

import math
def a(n): return (n/2**int(math.log(n - (n & n - 1), 2)) + 1)/2 # Indranil Ghosh, Apr 24 2017

def A003602(n): return (n>>(n&-n).bit_length())+1 # Chai Wah Wu, Jul 08 2022

(define (A003602 n) (let loop ((n n)) (if (even? n) (loop (/ n 2)) (/ (+ 1 n) 2)))) ;; Antti Karttunen, Feb 04 2015

a(n) = (A000265(n) + 1)/2.
a((2*k-1)*2^m) = k, for m >= 0 and k >= 1. - Robert G. Wilson v, May 23 2006
Inverse Weigh transform of A035528. - Christian G. Bower
G.f.: 1/x * Sum_{k>=0} x^2^k/(1-2*x^2^(k+1) + x^2^(k+2)). - Ralf Stephan, Jul 24 2003
a(2*n-1) = n and a(2*n) = a(n). - Pab Ter (pabrlos2(AT)yahoo.com), Oct 25 2005
a(A118413(n,k)) = A002024(n,k); = a(A118416(n,k)) = A002260(n,k); a(A014480(n)) = A001511(A014480(n)). - Reinhard Zumkeller, Apr 27 2006
Ordinal transform of A001511. - Franklin T. Adams-Watters, Aug 28 2006
a(n) = A249745(A126760(A003961(n))) = A249745(A253887(A048673(n))). That is, this sequence plays the same role for the numbers in array A135764 as A126760 does for the odd numbers in array A135765. - Antti Karttunen, Feb 04 2015 & Jan 19 2016
G.f. satisfies g(x) = g(x^2) + x/(1-x^2)^2. - Robert Israel, Apr 24 2015
a(n) = A181988(n)/A001511(n). - L. Edson Jeffery, Nov 21 2015
a(n) = A025480(n-1) + 1. - R. J. Mathar, May 19 2016
a(n) = A110963(2n-1) = A349135(4*n). - Antti Karttunen, Apr 18 2022
a(n) = (1 + n)/2, for n odd; a(n) = a(n/2), for n even. - David James Sycamore, Jul 28 2022
a(n) = n/2^A001511(n) + 1/2. - Alan Michael Gómez Calderón, Oct 06 2023
a(n) = A123390(A118319(n)). - Flávio V. Fernandes, Mar 02 2025

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 25 2005

A220466 a((2n-1)2^p) = 4^p(n-1) + 2^(p-1)(1+2^p), p >= 0 and n >= 1.

Original entry on oeis.org

1, 3, 2, 10, 3, 7, 4, 36, 5, 11, 6, 26, 7, 15, 8, 136, 9, 19, 10, 42, 11, 23, 12, 100, 13, 27, 14, 58, 15, 31, 16, 528, 17, 35, 18, 74, 19, 39, 20, 164, 21, 43, 22, 90, 23, 47, 24, 392, 25, 51, 26, 106, 27, 55, 28, 228, 29, 59, 30, 122, 31, 63, 32, 2080, 33, 67, 34, 138, 35

Offset: 1

Johannes W. Meijer, Dec 24 2012

The a(n) appeared in the analysis of A220002, a sequence related to the Catalan numbers.
The first Maple program makes use of a program by Peter Luschny for the calculation of the a(n) values. The second Maple program shows that this sequence has a beautiful internal structure, see the first formula, while the third Maple program makes optimal use of this internal structure for the fast calculation of a(n) values for large n.
The cross references lead to sequences that have the same internal structure as this sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Ralf Stephan, Some divide-and-conquer sequences with simple ordinary generating functions, The OEIS, Jan 01 2004.

Cf. A000027 (the natural numbers), A000120 (1's-counting sequence), A000265 (remove 2's from n), A001316 (Gould's sequence), A001511 (the ruler function), A003484 (Hurwitz-Radon numbers), A003602 (a fractal sequence), A006519 (highest power of 2 dividing n), A007814 (binary carry sequence), A010060 (Thue-Morse sequence), A014577 (dragon curve), A014707 (dragon curve), A025480 (nim-values), A026741, A035263 (first Feigenbaum symbolic sequence), A037227, A038712, A048460, A048896, A051176, A053381 (smooth nowhere-zero vector fields), A055975 (Gray code related), A059134, A060789, A060819, A065916, A082392, A085296, A086799, A088837, A089265, A090739, A091512, A091519, A096268, A100892, A103391, A105321 (a fractal sequence), A109168 (a continued fraction), A117973, A129760, A151930, A153733, A160467, A162728, A181988, A182241, A191488 (a companion to Gould's sequence), A193365, A220466 (this sequence).

-- Following Ralf Stephan's recurrence:
import Data.List (transpose)
a220466 n = a006519_list !! (n-1)
a220466_list = 1 : concat
   (transpose [zipWith (-) (map (* 4) a220466_list) a006519_list, [2..]])
-- Reinhard Zumkeller, Aug 31 2014

# First Maple program
a := n -> 2^padic[ordp](n, 2)*(n+1)/2 : seq(a(n), n=1..69); # Peter Luschny, Dec 24 2012
# Second Maple program
nmax:=69: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := 4^p*(n-1)  + 2^(p-1)*(1+2^p) od: od: seq(a(n), n=1..nmax);
# Third Maple program
nmax:=69: for p from 0 to ceil(simplify(log[2](nmax))) do n:=2^p: n1:=1: while n <= nmax do a(n) := 4^p*(n1-1)+2^(p-1)*(1+2^p): n:=n+2^(p+1): n1:= n1+1: od: od:  seq(a(n), n=1..nmax);

A220466 = Module[{n, p}, p = IntegerExponent[#, 2]; n = (#/2^p + 1)/2; 4^p*(n - 1) + 2^(p - 1)*(1 + 2^p)] &; Array[A220466, 50] (* JungHwan Min, Aug 22 2016 *)

a(n)=if(n%2,n\2+1,4*a(n/2)-2^valuation(n/2,2)) \\ Ralf Stephan, Dec 17 2013

a((2*n-1)*2^p) = 4^p*(n-1) + 2^(p-1)*(1+2^p), p >= 0 and n >= 1. Observe that a(2^p) = A007582(p).
a(n) = ((n+1)/2)*(A060818(n)/A060818(n-1))
a(n) = (-1/64)*(q(n+1)/q(n))/(2*n+1) with q(n) = (-1)^(n+1)*2^(4*n-5)*(2*n)!*A060818(n-1) or q(n) = (1/8)*A220002(n-1)*1/(A098597(2*n-1)/A046161(2*n))*1/(A008991(n-1)/A008992(n-1))
Recurrence: a(2n) = 4a(n) - 2^A007814(n), a(2n+1) = n+1. - Ralf Stephan, Dec 17 2013

A347957 Dirichlet convolution of A001221 (omega) with A003602 (Kimberling's paraphrases).

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 3, 3, 6, 1, 9, 1, 7, 7, 4, 1, 14, 1, 11, 8, 9, 1, 13, 4, 10, 8, 13, 1, 28, 1, 5, 10, 12, 9, 25, 1, 13, 11, 16, 1, 34, 1, 17, 22, 15, 1, 17, 5, 25, 13, 19, 1, 38, 11, 19, 14, 18, 1, 49, 1, 19, 26, 6, 12, 46, 1, 23, 16, 44, 1, 36, 1, 22, 31, 25, 12, 52, 1, 21, 22, 24, 1, 60, 14, 25, 19, 25, 1, 86

Offset: 1

Antti Karttunen, Sep 20 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Jon Maiga, Computer-generated formulas for A347957, Sequence Machine.

Cf. A001221, A003602, A023900, A181988, A205745.

Cf. also A328203, A347954, A347955, A347956.

A003602(n) = (1+(n>>valuation(n,2)))/2;
A347957(n) = sumdiv(n,d,omega(n/d)*A003602(d));

a(n) = Sum_{d|n} A001221(n/d) * A003602(d).
From Antti Karttunen, Nov 13 2021: (Start)
The following two convolutions were found by Jon Maiga's Sequence Machine search algorithm. The first one is obvious, and even the second one should not be too hard to prove:
a(n) = Sum_{d|n} A023900(n/d) * A347956(d).
a(n) = Sum_{d|n} A181988(n/d) * A205745(d).
(End)

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A349346 Dirichlet inverse of A181988, where A181988(n) = A001511(n)*A003602(n).

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A349347 Sum of A181988 and its Dirichlet inverse, where A181988(n) = A001511(n)*A003602(n).

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A001511 The ruler function: exponent of the highest power of 2 dividing 2n. Equivalently, the 2-adic valuation of 2n.

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A003602 Kimberling's paraphrases: if n = (2k-1)*2^m then a(n) = k.

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A220466 a((2*n-1)*2^p) = 4^p*(n-1) + 2^(p-1)*(1+2^p), p >= 0 and n >= 1.

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A347957 Dirichlet convolution of A001221 (omega) with A003602 (Kimberling's paraphrases).

A220466 a((2n-1)2^p) = 4^p(n-1) + 2^(p-1)(1+2^p), p >= 0 and n >= 1.