A220466 -id:A220466 - cp's OEIS frontend

A048896 a(n) = 2^(A000120(n+1) - 1), n >= 0.

1, 1, 2, 1, 2, 2, 4, 1, 2, 2, 4, 2, 4, 4, 8, 1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, 1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, 2, 4, 4, 8, 4, 8, 8, 16, 4, 8, 8, 16, 8, 16, 16, 32, 1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, 2, 4, 4, 8, 4, 8, 8, 16, 4, 8, 8, 16, 8, 16, 16, 32, 2, 4, 4

Offset: 0

Wolfdieter Lang

a(n) = 2^A048881 = 2^{maximal power of 2 dividing the n-th Catalan number (A000108)}. [Comment corrected by N. J. A. Sloane, Apr 30 2018]
Row sums of triangle A128937. - Philippe Deléham, May 02 2007
a(n) = sum of (n+1)-th row terms of triangle A167364. - Gary W. Adamson, Nov 01 2009
a(n), n >= 1: Numerators of Maclaurin series for 1 - ((sin x)/x)^2, A117972(n), n >= 2: Denominators of Maclaurin series for 1 - ((sin x)/x)^2, the correlation function in Montgomery's pair correlation conjecture. - Daniel Forgues, Oct 16 2011
For n > 0: a(n) = A007954(A007931(n)). - Reinhard Zumkeller, Oct 26 2012
a(n) = A261363(2*(n+1), n+1). - Reinhard Zumkeller, Aug 16 2015
From Gus Wiseman, Oct 30 2022: (Start)
Also the number of coarsenings of the (n+1)-th composition in standard order. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. See link for sequences related to standard compositions. For example, the a(10) = 4 coarsenings of (2,1,1) are: (2,1,1), (2,2), (3,1), (4).
Also the number of times n+1 appears in A357134. For example, 11 appears at positions 11, 20, 33, and 1024, so a(10) = 4.
(End)

			From _Omar E. Pol_, Jul 21 2009: (Start)
If written as a triangle:
  1;
  1,2;
  1,2,2,4;
  1,2,2,4,2,4,4,8;
  1,2,2,4,2,4,4,8,2,4,4,8,4,8,8,16;
  1,2,2,4,2,4,4,8,2,4,4,8,4,8,8,16,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32;
  ...,
the first half-rows converge to Gould's sequence A001316.
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Neil J. Calkin, Eunice Y. S. Chan, and Robert M. Corless, Some Facts and Conjectures about Mandelbrot Polynomials, Maple Transactions (2021) Vol. 1, No. 1 Article 1.
Neil J. Calkin, Eunice Y. S. Chan, Robert M. Corless, David J. Jeffrey, and Piers W. Lawrence, A Fractal Eigenvector, arXiv:2104.01116 [math.DS], 2021.
Emeric Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, arXiv:math/0407326 [math.CO], 2004; J. Num. Theory 117 (2006), 191-215.
OEIS Wiki, Montgomery's pair correlation conjecture
Gus Wiseman, Statistics, classes, and transformations of standard compositions

This is Guy Steele's sequence GS(3, 5) (see A135416).
Equals first right hand column of triangle A160468.
Equals A160469(n+1)/A002425(n+1).
Cf. A000079, A001316, A117972, A160476, A167364, A220466, A261363.
Standard compositions are listed by A066099.
The opposite version (counting refinements) is A080100.
The version for Heinz numbers of partitions is A317141.
Cf. A000120, A001511, A029931, A048793, A058891, A070939, A272919, A357134.

a048896 n = a048896_list !! n
a048896_list = f [1] where f (x:xs) = x : f (xs ++ [x,2*x])
-- Reinhard Zumkeller, Mar 07 2011

import Data.List (transpose)
a048896 = a000079 . a000120
a048896_list = 1 : concat (transpose
   [zipWith (-) (map (* 2) a048896_list) a048896_list,
    map (* 2) a048896_list])
-- Reinhard Zumkeller, Jun 16 2013

[Numerator(2^n / Factorial(n+1)): n in [0..100]]; // Vincenzo Librandi, Apr 12 2014

a := n -> 2^(add(i,i=convert(n+1,base,2))-1): seq(a(n), n=0..97); # Peter Luschny, May 01 2009

NestList[Flatten[#1 /. a_Integer -> {a, 2 a}] &, {1}, 4] // Flatten (* Robert G. Wilson v, Aug 01 2012 *)
Table[Numerator[2^n / (n + 1)!], {n, 0, 200}] (* Vincenzo Librandi, Apr 12 2014 *)
Denominator[Table[BernoulliB[2*n] / (Zeta[2*n]/Pi^(2*n)), {n, 1, 100}]] (* Terry D. Grant, May 29 2017 *)
Table[Denominator[((2 n)!/2^(2 n + 1)) (-1)^n], {n, 1, 100}]/4 (* Terry D. Grant, May 29 2017 *)
2^IntegerExponent[CatalanNumber[Range[0,100]],2] (* Harvey P. Dale, Apr 30 2018 *)

a(n)=if(n<1,1,if(n%2,a(n/2-1/2),2*a(n-1)))

a(n) = 1 << (hammingweight(n+1)-1); \\ Kevin Ryde, Feb 19 2022

a(n) = 2^A048881(n).
a(n) = 2^k if 2^k divides A000108(n) but 2^(k+1) does not divide A000108(n).
It appears that a(n) = Sum_{k=0..n} binomial(2*(n+1), k) mod 2. - Christopher Lenard (c.lenard(AT)bendigo.latrobe.edu.au), Aug 20 2001
a(0) = 1; a(2*n) = 2*a(2*n-1); a(2*n+1) = a(n).
a(n) = (1/2) * A001316(n+1). - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 26 2004
It appears that a(n) = Sum_{k=0..2n} floor(binomial(2n+2, k+1)/2)(-1)^k = 2^n - Sum_{k=0..n+1} floor(binomial(n+1, k)/2). - Paul Barry, Dec 24 2004
a(n) = Sum_{k=0..n} (T(n,k) mod 2) where T = A039598, A053121, A052179, A124575, A126075, A126093. - Philippe Deléham, May 02 2007
a(n) = numerator(b(n)), where sin(x)^2/x = Sum_{n>0} b(n)*(-1)^n x^(2*n-1). - Vladimir Kruchinin, Feb 06 2013
a((2*n+1)*2^p-1) = A001316(n), p >= 0 and n >= 0. - Johannes W. Meijer, Feb 12 2013
a(n) = numerator(2^n / (n+1)!). - Vincenzo Librandi, Apr 12 2014
a(2n) = (2n+1)!/(n!n!)/A001803(n). - Richard Turk, Aug 23 2017
a(2n-1) = (2n-1)!/(n!(n-1)!)/A001790(n). - Richard Turk, Aug 23 2017

New definition from N. J. A. Sloane, Mar 01 2008

A096268 Period-doubling sequence (or period-doubling word): fixed point of the morphism 0 -> 01, 1 -> 00.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jun 22 2004

nonn

Take highest power of 2 dividing n (A007814(n+1)), read modulo 2.
For the scale-invariance properties see Hendriks et al., 2012.
This is the sequence that results from the ternary Thue-Morse sequence (A036577) if all twos in that sequence are replaced by zeros. - Nathan Fox, Mar 12 2013
This sequence can be used to draw the Von Koch snowflake with a suitable walk in the plane. Start from the origin then the n-th step is "turn +Pi/3 if a(n)=0 and turn -2*Pi/3 if a(n)=1" (see link for a plot of the first 200000 steps). - Benoit Cloitre, Nov 10 2013
1 iff the number of trailing zeros in the binary representation of n+1 is odd. - Ralf Stephan, Nov 11 2013
Equivalently, with offset 1, the characteristic function of A036554 and an indicator for the A003159/A036554 classification of positive integers. - Peter Munn, Jun 02 2020

			Start: 0
Rules:
  0 --> 01
  1 --> 00
-------------
0:   (#=1)
  0
1:   (#=2)
  01
2:   (#=4)
  0100
3:   (#=8)
  01000101
4:   (#=16)
  0100010101000100
5:   (#=32)
  01000101010001000100010101000101
6:   (#=64)
  0100010101000100010001010100010101000101010001000100010101000100
7:   (#=128)
  010001010100010001000101010001010100010101000100010001010100010001000101010...
[_Joerg Arndt_, Jul 06 2011]

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, Table of n, a(n) for n = 0..10000 (first 1022 terms from T. D. Noe)
Jean-Paul Allouche, Michael Baake, Julien Cassaigne, and David Damanik, Palindrome complexity, arXiv:math/0106121 [math.CO], 2001; Theoretical Computer Science, 292 (2003), 9-31.
Scott Balchin and Dan Rust, Computations for Symbolic Substitutions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.4.1.
Benoit Cloitre, 200000 steps in the plane using "turn +Pi/3 if a(n)=0 and -2Pi/3 otherwise".
Cristian Cobeli, Mihai Prunescu, and Alexandru Zaharescu, On non-holonomicity, transcendence and p-adic valuations, arXiv:2412.16517 [math.NT], 2024. See p. 12.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
G. Jörg Endrullis, Dimitri Hendriks, and Jan Willem Klop, Degrees of streams.
Robbert Fokkink and Gandhar Joshi, Anti-recurrence sequences, arXiv:2506.13337 [math.NT], 2025. See pp. 2, 18.
Dimitri Hendriks, Frits G. W. Dannenberg, Jorg Endrullis, Mark Dow and Jan Willem Klop, Arithmetic Self-Similarity of Infinite Sequences, arXiv 1201.3786 [math.CO], 2012.
Andreas M. Hinz, Sandi Klavžar, Uroš Milutinović, and Ciril Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 79. Book's website
Shuo Li, Palindromic length sequence of the ruler sequence and of the period-doubling sequence, arXiv:2007.08317 [math.CO], 2020.
Laszlo Mérai and A. Winterhof, On the Nth linear complexity of automatic sequences, arXiv preprint arXiv:1711.10764 [math.NT], 2017.
Jeong-Yup Lee, D. Flom and S. I. Ben-Abraham, Multidimensional period doubling structures, Acta Crystallographica Section A: Foundations, (2016). A72, 391-394.
Aline Parreau, Michel Rigo, Eric Rowland, and Elise Vandomme, A new approach to the 2-regularity of the l-abelian complexity of 2-automatic sequences, arXiv:1405.3532 [cs.FL], 2014-2015.
Saúl Pilatowsky-Cameo, Soonwon Choi, and Wen Wei Ho, Critically slow Hilbert-space ergodicity in quantum morphic drives, arXiv:2502.06936 [quant-ph], 2025. See pp. 6, 15.
Narad Rampersad and Manon Stipulanti, The Formal Inverse of the Period-Doubling Sequence, arXiv:1807.11899 [math.CO], 2018.
Luke Schaeffer and Jeffrey Shallit, Closed, Palindromic, Rich, Privileged, Trapezoidal, and Balanced Words in Automatic Sequences, Electronic Journal of Combinatorics 23(1) (2016), Article P1.25.
Index entries for sequences that are fixed points of mappings.
Index entries for sequences related to binary expansion of n.

Not the same as A073059!
Swapping 0 and 1 gives A035263.
Cf. A096269, A096270, A071858, A220466, A036577.
Cf. A056832, A123087 (partial sums).
With offset 1, classification indicator for A003159/A036554.
Also with offset 1: A007814 mod 2 (cf. A096271 for mod 3), A048675 mod 2 (cf. A332813 for mod 3), A059975 mod 2.

a096268 = (subtract 1) . a056832 . (+ 1)
-- Reinhard Zumkeller, Jul 29 2014

[Valuation(n+1, 2) mod 2: n in [0..100]]; // Vincenzo Librandi, Jul 20 2016

nmax:=104: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 0 to ceil(nmax/(p+2))+1 do a((2*n+1)*2^p-1) := p mod 2 od: od: seq(a(n), n=0..nmax); # Johannes W. Meijer, Feb 02 2013
# second Maple program:
a:= proc(n) a(n):= `if`(n::even, 0, 1-a((n-1)/2)) end:
seq(a(n), n=0..125);  # Alois P. Heinz, Mar 20 2019

Nest[ Flatten[ # /. {0 -> {1, 0}, 1 -> {0, 0}}] &, {1}, 7] (* Robert G. Wilson v, Mar 05 2005 *)
{{0}}~Join~SubstitutionSystem[{0 -> {0, 1}, 1 -> {0, 0}}, {1}, 6] // Flatten (* Michael De Vlieger, Aug 15 2016 *)

a(n)=valuation(n+1,2)%2 \\ Ralf Stephan, Nov 11 2013

def A096268(n): return (~(n+1)&n).bit_length()&1 # Chai Wah Wu, Jan 09 2023

Recurrence: a(2*n) = 0, a(4*n+1) = 1, a(4*n+3) = a(n). - Ralf Stephan, Dec 11 2004
The recurrence may be extended backwards, with a(-1) = 1. - S. I. Ben-Abraham, Apr 01 2013
a(n) = 1 - A035263(n-1). - Reinhard Zumkeller, Aug 16 2006
Dirichlet g.f.: zeta(s)/(1+2^s). - Ralf Stephan, Jun 17 2007
Let T(x) be the g.f., then T(x) + T(x^2) = x^2/(1-x^2). - Joerg Arndt, May 11 2010
Let 2^k||n+1. Then a(n)=1 if k is odd, a(n)=0 if k is even. - Vladimir Shevelev, Aug 25 2010
a(n) = A007814(n+1) mod 2. - Robert G. Wilson v, Jan 18 2012
a((2*n+1)*2^p-1) = p mod 2, p >= 0 and n >= 0. - Johannes W. Meijer, Feb 02 2013
a(n) = A056832(n+1) - 1. - Reinhard Zumkeller, Jul 29 2014
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/3. = Amiram Eldar, Sep 18 2022

Corrected by Jeremy Gardiner, Dec 12 2004

More terms from Robert G. Wilson v, Feb 26 2005

A060789 a(n) = n / (gcd(n,2) * gcd(n,3)).

Original entry on oeis.org

1, 1, 1, 2, 5, 1, 7, 4, 3, 5, 11, 2, 13, 7, 5, 8, 17, 3, 19, 10, 7, 11, 23, 4, 25, 13, 9, 14, 29, 5, 31, 16, 11, 17, 35, 6, 37, 19, 13, 20, 41, 7, 43, 22, 15, 23, 47, 8, 49, 25, 17, 26, 53, 9, 55, 28, 19, 29, 59, 10, 61, 31, 21, 32, 65, 11, 67, 34, 23, 35, 71, 12, 73, 37, 25, 38, 77

Offset: 1

Len Smiley, Apr 26 2001

a(n+2) is absolute value of numerator of determinant of n X n matrix with M(i,j) = 2/(i(i+1)) if i=j otherwise 1. - Alexander Adamchuk, May 19 2006
Numerator of n/(n+6). - Gerry Martens, Aug 06 2015
In addition to being multiplicative, this sequence is also a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n, m)) for n, m >= 1. In particular, it follows that a(n) is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, Feb 20 2019
Connected to the Diophantine equation x^2 + (x + 1)^2 + … + (x + k)^2 = C, C an integer, x >= 0, k >= 0. Rewritten it is (k + 1)*x^2 + k*(k+1)*x + k*(k + 1)*(2*k + 1)/6 = C. The existence of its solutions depends on gcd(k + 1, k*(k + 1), k*(k + 1)*(2*k + 1)/6). - Ctibor O. Zizka, Oct 04 2023

Harry J. Smith, Table of n, a(n) for n=1..1000
Wikipedia, Quasi-polynomial
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,2,0,0,0,0,0,-1).

Cf. A011782, A109047, A146535, A220466.

Cf. other sequences given by the formula n/gcd(n,k) = numerator(n/(n + k)): A026741 (k = 2), A051176 (k = 3), A060819 (k = 4), A060791 (k = 5), A106608 thru A106612 (k = 7 thru 11), A051724 (k = 12), A106614 thru A106621 (k = 13 thru 20).

List([1..80],n->n/(Gcd(n,2)*Gcd(n,3))); # Muniru A Asiru, Feb 20 2019

[n/(Gcd(n,2)*Gcd(n,3)) : n in [1..100]]; // Wesley Ivan Hurt, Aug 06 2015

I:=[1,1,1,2,5,1,7,4,3,5,11,2]; [n le 12 select I[n] else 2*Self(n-6)-Self(n-12): n in [1..80]]; // Vincenzo Librandi, Aug 07 2015

a := proc(n): if n = 1 then 1 else denom((4*n-6)/n) fi: end: seq(a(n), n=1..77); # Johannes W. Meijer, Dec 19 2012

Numerator[Table[(-1)^(n+1) Det[ DiagonalMatrix[ Table[ 2/(i(i+1)) - 1, {i, 1, n-2} ] ] + 1 ], {n, 30} ]] (* Alexander Adamchuk, May 19 2006 *)
Table[Numerator[(n+3)/(n+2)/(n+1)/n],{n,60}] (* Vladimir Joseph Stephan Orlovsky, Nov 17 2009 *)
Table[n/(GCD[n, 2] GCD[n, 3]), {n, 100}] (* Wesley Ivan Hurt, Aug 06 2015 *)
LinearRecurrence[{0,0,0,0,0,2,0,0,0,0,0,-1}, {1,1,1,2,5,1,7,4,3,5,11,2}, 80] (* Vincenzo Librandi, Aug 07 2015 *)

a(n) = { n / (gcd(n, 2) * gcd(n, 3)) } \\ Harry J. Smith, Jul 11 2009

[lcm(n,6)/6 for n in range(1, 78)] # Zerinvary Lajos, Jun 07 2009

G.f.: x*(1 + x + x^2 + 2*x^3 + 5*x^4 + x^5 + 5*x^6 + 2*x^7 + x^8 + x^9 + x^10)/(1 - x^6)^2.
Multiplicative with a(2^e)=2^(e-1), a(3^e)=3^(e-1), a(p^e)=p^e, p>3. - Vladeta Jovovic, Sep 09 2004
a(n) = Numerator[(-1)^(n+1)*Det[DiagonalMatrix[Table[2/(i(i+1))-1, {i,1,n-2}]]+1]], n>2. - Alexander Adamchuk, May 19 2006
a(n) divides n. a(6k) = k for integer k>0. a(p^k) = p^k for prime p>3 and integer k>0. - Alexander Adamchuk, Sep 20 2006
From R. J. Mathar, Apr 18 2011: (Start)
a(n) = A109047(n)/6.
Dirichlet g.f. zeta(s-1)*(1-1/2^s-2/3^s+2/6^s). (End)
a(n) = denominator((4*n-6)/n), n >= 2, with a(1) = 1. - Johannes W. Meijer, Dec 19 2012
a((2*n-1)*2^p) = A011782(p)*A146535(n), p >= 0. - Johannes W. Meijer, Feb 06 2013
a(n) = 2*a(n-6) - a(n-12) for n >= 12. - Robert Israel, Aug 06 2015
a(n) = gcd((n-1)*n*(n+1)/6, n). - Lechoslaw Ratajczak, Feb 16 2017
From Peter Bala, Feb 13 2019: (Start)
a(n) = n/gcd(n,n + 6) = n/gcd(n,6).
a(n) = n/b(n), where b(n) is the purely periodic sequence [1,2,3,2,1,6,...] with period 6.
a(n) is a quasi-polynomial in n: a(6*n+1) = 6*n + 1; a(6*n+2) = 3*n + 1; a(6*n+3) = 2*n + 1; a(6*n+4) = 3*n + 2; a(6*n+5) = 6*n + 5; a(6*n) = n.
a(n) = numerator(n/(n + 6)); a(n) = denominator((n + 6)/n).
(End)
Sum_{k=1..n} a(k) ~ 7*n^2/24. - Vaclav Kotesovec, Aug 09 2022
From Ctibor O. Zizka, Oct 04 2023: (Start)
For k >=0, a(k) = gcd(k + 1, k*(k + 1), k*(k + 1)*(2*k + 1)/6).
If (k mod 6) = 0 or 4 then a(k) = (k + 1).
If (k mod 6) = 1 or 3 then a(k) = (k + 1)/2.
If (k mod 6) = 2 then a(k) = (k + 1)/3.
If (k mod 6) = 5 then a(k) = (k + 1)/6. (End)

A129760 Bitwise AND of binary representation of n-1 and n.

Original entry on oeis.org

0, 0, 2, 0, 4, 4, 6, 0, 8, 8, 10, 8, 12, 12, 14, 0, 16, 16, 18, 16, 20, 20, 22, 16, 24, 24, 26, 24, 28, 28, 30, 0, 32, 32, 34, 32, 36, 36, 38, 32, 40, 40, 42, 40, 44, 44, 46, 32, 48, 48, 50, 48, 52, 52, 54, 48, 56, 56, 58, 56, 60, 60, 62, 0, 64, 64, 66, 64, 68, 68, 70, 64, 72, 72, 74

Offset: 1

Russ Cox, May 15 2007

Also the number of Ducci sequences with period n.
Also largest number less than n having in binary representation fewer ones than n has; A048881(n-1) = A000120(a(n)) = A000120(n)-1. - Reinhard Zumkeller, Jun 30 2010
a(n) is the parent of vertex n in the binomial tree.  The binomial tree is root vertex n=0, then for n>=1 the parent of n is n with its least significant 1-bit changed to a 0-bit.  Binomial tree order 5, n=0 to 31 inclusive, is the frontispiece of Knuth volume 1, second and subsequent editions.  Vertices are shown there with n in binary dots and a(n) is the next vertex towards the root at the bottom of the page. - Kevin Ryde, Jul 24 2019

			a(6) = 6 AND 5 = binary 110 AND 101 = binary 100 = 4.

Donald E. Knuth, The Art of Computer Programming, volume 1, second edition, frontispiece. Reproduced with brief description of the art in Donald E. Knuth, Selected Papers on Fun and Games, 2010, Chapter 47 Geek Art, figure 16, page 679.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Ron Brown and Jonathan L. Merzel, The number of Ducci sequences with a given period, Fib. Quart., 45 (2007), 115-121.

Cf. A038712, A086799, A104594, A059991, A006519, A109168, A220466.

C
```
int a(int n) { return n & (n-1); }
```

[n - 2^Valuation(n, 2): n in [1..100]]; // Vincenzo Librandi, Jul 25 2019

nmax := 75: 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) := (2*n-2) * 2^p od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Jun 22 2011, revised Jan 25 2013
A129760 := n -> Bits:-And(n-1, n):
seq(A129760(n), n=1..75); # Peter Luschny, Sep 26 2019

Table[BitAnd[n, n - 1], {n, 1, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 19 2011 *)

a(n)=bitand(n,n-1) \\ Charles R Greathouse IV, Jun 23 2011

def a(n): return n & (n-1)
print([a(n) for n in range(1, 71)]) # Michael S. Branicky, Jul 13 2022

a(n) = n AND n-1.
Equals n - A006519(n). - N. J. A. Sloane, May 26 2008
From Johannes W. Meijer, Jun 22 2011: (Start)
a((2*n-1)*2^p) = (2*n-2)*(2^p), p>=0.
a(2*n-1) = (2*n-2), n>=1, and a(2^p+1) = 2^p, p>=1. (End)

A014707 a(4n) = 0, a(4n+2) = 1, a(2n+1) = a(n).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

The regular paper-folding (or dragon curve) sequence.
It appears that the sequence of run lengths is A088431. - Dimitri Hendriks, May 06 2010
Runs of three consecutive ones appear around positions n = 22, 46, 54, 86, 94, 118, 150, 174, 182, ..., or for n of the form 2^(k+3)*(4*t+3)-2, k >= 0, t >= 0. - Vladimir Shevelev, Mar 19 2011

Guy Melançon, Factorizing infinite words using Maple, MapleTech journal, Vol. 4, No. 1, 1997, pp. 34-42, esp. p. 36.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Jean-Paul Allouche, Anna Lubiw, Michel France, Alfred van der Poorten, and Jeffrey Shallit, Convergents of folded continued fractions, Acta Arithmetica 77 (1996), 77-96.
Cristina Ballantine and George Beck, Partitions enumerated by self-similar sequences, arXiv:2303.11493 [math.CO], 2023.
Paul Barry, On the Gap-sum and Gap-product Sequences of Integer Sequences, arXiv:2104.05593 [math.CO], 2021.
Jörg Endrullis, Dimitri Hendriks, and Jan Willem Klop, Degrees of streams, see table 1 "PF".
Jui-Yi Kao, Narad Rampersad, Jeffrey Shallit, and Manuel Silva, Words avoiding repetitions in arithmetic progressions, Theoretical Computer Science, Vol. 391, No. 1-2 (2008), pp. 126-137; arXiv preprint, arXiv:math/0608607 [math.CO], 2006.
Guy Melançon, Lyndon factorization of infinite words, STACS 96 (Grenoble, 1996), 147-154, Lecture Notes in Comput. Sci., 1046, Springer, Berlin, 1996. Math. Rev. 98h:68188.
Jeffrey Shallit, Cloitre's Self-Generating Sequence, arXiv:2501.00784 [math.CO], 2025.
László Tóth, Evaluating zeta(s) At Odd Positive Integers Using Automatic Dirichlet Series, arXiv:2508.04151 [math.NT], 2025.
Index entries for sequences obtained by enumerating foldings.

Equals 1 - A014577, which see for further references. Also a(n) = A038189(n+1).
The following are all essentially the same sequence: A014577, A014707, A014709, A014710, A034947, A038189, A082410.
Cf. A220466

a014707 n = a014707_list !! n
a014707_list = f 0 $ cycle [0,0,1,0] where
   f i (x:_:xs) = x : a014707 i : f (i+1) xs
-- Reinhard Zumkeller, Sep 28 2011

nmax:=92: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 0 to ceil(nmax/(p+2))+1 do a((2*n+1)*2^p-1) := n mod 2 od: od: seq(a(n), n=0..nmax); # Johannes W. Meijer, Jan 28 2013
# second Maple program:
a:= proc(n) option remember;
     `if`(n::even, irem(n/2, 2), a((n-1)/2))
    end:
seq(a(n), n=0..92);  # Alois P. Heinz, Jun 27 2022

a[n_ /; Mod[n, 4] == 0] = 0; a[n_ /; Mod[n, 4] == 2] = 1; a[n_ /; Mod[n, 2] == 1] := a[n] = a[(n - 1)/2]; Table[a[n],{n,0,92}] (* Jean-François Alcover, May 17 2011 *)
(1 - JacobiSymbol[-1, Range[100]])/2 (* Paolo Xausa, May 26 2024 *)

a(n)=n+=1;my(h=bitand(n,-n));n=bitand(n,h<<1);n!=0; \\ Joerg Arndt, Apr 09 2021

def A014707(n):
    s = bin(n+1)[2:]
    m = len(s)
    i = s[::-1].find('1')
    return int(s[m-i-2]) if m-i-2 >= 0 else 0 # Chai Wah Wu, Apr 08 2021

def A014707(n): n+=1; h=n&-n; n=n&(h<<1); return int(n!=0)
print([A014707(n) for n in range(93)]) # Michael S. Branicky, Mar 29 2024 after Joerg Arndt

a(A091072(n)-1) = 0; a(A091067(n)-1) = 1. - Reinhard Zumkeller, Sep 28 2011 [Adjusted to match offset by Peter Munn, Jul 01 2022]
a(n) = (1-Jacobi(-1,n+1))/2 (cf. A034947). - N. J. A. Sloane, Jul 27 2012 [Adjusted to match offset by Peter Munn, Jul 01 2022]
Set a=0, b=1, S(0)=a, S(n+1) = S(n)aF(S(n)), where F(x) reverses x and then interchanges a and b; sequence is limit S(infinity).
a((2*n+1)*2^p-1) = n mod 2, p >= 0. - Johannes W. Meijer, Jan 28 2013
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/2. - Amiram Eldar, Aug 31 2024

More terms from Scott C. Lindhurst (ScottL(AT)alumni.princeton.edu)

A103391 "Even" fractal sequence for the natural numbers: Deleting every even-indexed term results in the same sequence.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 3, 5, 2, 6, 4, 7, 3, 8, 5, 9, 2, 10, 6, 11, 4, 12, 7, 13, 3, 14, 8, 15, 5, 16, 9, 17, 2, 18, 10, 19, 6, 20, 11, 21, 4, 22, 12, 23, 7, 24, 13, 25, 3, 26, 14, 27, 8, 28, 15, 29, 5, 30, 16, 31, 9, 32, 17, 33, 2, 34, 18, 35, 10, 36, 19, 37, 6, 38, 20, 39, 11, 40, 21, 41, 4, 42, 22, 43, 12, 44, 23, 45, 7, 46, 24, 47, 13, 48, 25, 49, 3, 50, 26, 51, 14, 52, 27, 53, 8

Offset: 1

Eric Rowland, Mar 20 2005

A003602 is the "odd" fractal sequence for the natural numbers.

Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(A005940(i)) = A348717(A005940(j)) for all i, j >= 1. A365718 is an analogous sequence related to A356867 (Doudna variant D(3)). - Antti Karttunen, Sep 17 2023

Antti Karttunen, Table of n, a(n) for n = 1..65537 (terms 1..10000 from Reinhard Zumkeller)

Cf. A003602, A005940, A025480, A220466, A286387, A353368 (Dirichlet inverse).
Cf. also A110962, A110963, A365718.
Differs from A331743(n-1) for the first time at n=192, where a(192) = 97, while A331743(191) = 23.
Differs from A351460.

-- import Data.List (transpose)
a103391 n = a103391_list !! (n-1)
a103391_list = 1 : ks where
   ks = concat $ transpose [[2..], ks]
-- Reinhard Zumkeller, May 23 2013

nmax := 82: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 2 to ceil(nmax/(p+2))+1 do a((2*n-3)*2^p+1) := n od: od: a(1) := 1: seq(a(n), n=1..nmax); # Johannes W. Meijer, Jan 28 2013

a[n_] := ((n-1)/2^IntegerExponent[n-1, 2] + 3)/2; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 24 2023 *)

A003602(n) = (n/2^valuation(n, 2)+1)/2; \\ From A003602
A103391(n) = if(1==n,1,(1+A003602(n-1))); \\ Antti Karttunen, Feb 05 2020

def v(n): b = bin(n); return len(b) - len(b.rstrip("0"))
def b(n): return (n//2**v(n)+1)//2
def a(n): return 1 if n == 1 else 1 + b(n-1)
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, May 29 2022

def A103391(n): return (n-1>>(n-1&-n+1).bit_length())+2 if n>1 else 1 # Chai Wah Wu, Jan 04 2024

For n > 1, a(n) = A003602(n-1) + 1. - Benoit Cloitre, May 26 2007, indexing corrected by Antti Karttunen, Feb 05 2020
a((2*n-3)*2^p+1) = n, p >= 0 and n >= 2, with a(1) = 1. - Johannes W. Meijer, Jan 28 2013
Sum_{k=1..n} a(k) ~ n^2/6. - Amiram Eldar, Sep 24 2023

Data section extended up to a(105) (to better differentiate from several nearby sequences) by Antti Karttunen, Feb 05 2020

A037227 If n = 2^mk, k odd, then a(n) = 2m+1.

Original entry on oeis.org

1, 3, 1, 5, 1, 3, 1, 7, 1, 3, 1, 5, 1, 3, 1, 9, 1, 3, 1, 5, 1, 3, 1, 7, 1, 3, 1, 5, 1, 3, 1, 11, 1, 3, 1, 5, 1, 3, 1, 7, 1, 3, 1, 5, 1, 3, 1, 9, 1, 3, 1, 5, 1, 3, 1, 7, 1, 3, 1, 5, 1, 3, 1, 13, 1, 3, 1, 5, 1, 3, 1, 7, 1, 3, 1, 5, 1, 3, 1, 9, 1, 3, 1, 5, 1, 3, 1, 7, 1, 3, 1, 5, 1, 3, 1, 11, 1, 3, 1, 5, 1, 3

Offset: 1

N. J. A. Sloane

Take the number of rightmost zeros in the binary expansion of n, double it, and increment it by 1. - Ralf Stephan, Aug 22 2013

Gives the maximum possible number of n X n complex Hermitian matrices with the property that all of their nonzero real linear combinations are nonsingular (see Adams et al. reference). - Nathaniel Johnston, Dec 11 2013

T. D. Noe, Table of n, a(n) for n=1..1024
J. F. Adams, P. D. Lax, and R. S. Phillips, On matrices whose real linear combinations are nonsingular, Proceedings of the American Mathematical Society, 16:318-322, 1965.
D. B. Shapiro, Problem 10456: Anticommuting Matrices, Amer. Math. Monthly, 105 (1998), 565-566.
Index entries for sequences related to binary expansion of n

Cf. A001511, A002487, A007814, A005408, A016825, A017113, A075300, A051062, A220466.

a037227 = (+ 1) . (* 2) . a007814  -- Reinhard Zumkeller, Jun 30 2012

[2*Valuation(n, 2)+1: n in [1..120]]; // Vincenzo Librandi, Jun 19 2019

nmax:=102: 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):= 2*p+1: od: od: seq(a(n), n=1..nmax);  # Johannes W. Meijer, Feb 07 2013

a[n_] := Sum[(-1)^(d+1)*MoebiusMu[d]*DivisorSigma[0, n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 102}] (* Jean-François Alcover, Dec 31 2012, after Vladeta Jovovic *)
f[n_]:=Module[{z=Last[Split[IntegerDigits[n,2]]]},If[Union[z]={0},2* Length[ z]+1,1]]; Array[f,110] (* Harvey P. Dale, Jun 16 2019, after Ralf Stephan *)
Table[2 IntegerExponent[n, 2] + 1, {n, 120}] (* Vincenzo Librandi, Jun 19 2019 *)

a(n)=2*valuation(n,2)+1 \\ Charles R Greathouse IV, May 21 2015

def A037227(n): return ((~n & n-1).bit_length()<<1)+1 # Chai Wah Wu, Jul 05 2022

maxrow <- 6 # by choice
a <- 1
for(m in 0:maxrow){
for(k in 0:(2^m-1)) {
   a[2^(m+1)    +k] <- a[2^m+k]
   a[2^(m+1)+2^m+k] <- a[2^m+k]
}
   a[2^(m+1)      ] <- a[2^(m+1)] + 2
}
a
# Yosu Yurramendi, May 21 2015

a(n) = Sum_{d divides n} (-1)^(d+1)*mu(d)*tau(n/d). Multiplicative with a(p^e) = 2*e+1 if p = 2; 1 if p > 2. - Vladeta Jovovic, Apr 27 2003
a(n) = a(n-1)+(-1)^n*(a(floor(n/2))+1). - Vladeta Jovovic, Apr 27 2003
a(2*n) = a(n) + 2, a(2*n+1) = 1. a(n) = 2*A007814(n) + 1. - Ralf Stephan, Oct 07 2003
a(A005408(n)) = 1; a(A016825(n)) = 3; A017113(a(n)) = 5; A051062(a(n)) = 7. - Reinhard Zumkeller, Jun 30 2012
a((2*n-1)*2^p)  = 2*p+1, p  >= 0 and n >= 1. - Johannes W. Meijer, Feb 07 2013
From Peter Bala, Feb 07 2016: (Start)
a(n) = ( A002487(n-1) + A002487(n+1) )/A002487(n).
a(n*2^(k+1) + 2^k) = 2*k + 1 for n,k >= 0; thus a(2*n+1) = 1, a(4*n+2) = 3, a(8*n+4) = 5, a(16*n+8) = 7 and so on. Note the square array ( n*2^(k+1) + 2^k - 1 )n, k>=0 is the transpose of A075300.
G.f.: Sum_{n >= 0} (2*n + 1)*x^(2^n)/(1 - x^(2^(n+1))). (End)
a(n) = 2*floor(A002487(n-1)/A002487(n))+1 for n > 1. - I. V. Serov, Jun 15 2017
From Amiram Eldar, Nov 29 2022: (Start)
Dirichlet g.f.: zeta(s)*(2^s+1)/(2^s-1).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3. (End)

More terms from Erich Friedman

A086799 Replace all trailing 0's with 1's in binary representation of n.

Original entry on oeis.org

1, 3, 3, 7, 5, 7, 7, 15, 9, 11, 11, 15, 13, 15, 15, 31, 17, 19, 19, 23, 21, 23, 23, 31, 25, 27, 27, 31, 29, 31, 31, 63, 33, 35, 35, 39, 37, 39, 39, 47, 41, 43, 43, 47, 45, 47, 47, 63, 49, 51, 51, 55, 53, 55, 55, 63, 57, 59, 59, 63, 61, 63, 63, 127, 65, 67, 67, 71, 69, 71

Offset: 1

Reinhard Zumkeller, Aug 05 2003

a(k+1) = smallest number greater than k having in its binary representation exactly one 1 more than k has; A000120(a(n)) = A063787(n). - Reinhard Zumkeller, Jul 31 2010
a(n) is the least m >= n-1 such that the Hamming distance D(n-1,m) = 1. - Vladimir Shevelev, Apr 18 2012
The number of appearances of k equals the 2-adic valuation of k+1. - Ali Sada, Dec 20 2024

			a(20) = a('10100') = '10100' + '11' = '10111' = 23.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Eric Weisstein's World of Mathematics, Binary Carry Sequence
Eric Weisstein's World of Mathematics, Odd Part
Index entries for sequences related to binary expansion of n

Cf. A000051, A000079, A000120, A000265, A001620, A006519, A007814, A023416, A038712, A063787, A086784.

Cf. also A007088, A179857, A220466.

int a(int n) { return n | (n-1); } // Russ Cox, May 15 2007

a086799 n | even n    = (a086799 $ div n 2) * 2 + 1
          | otherwise = n
-- Reinhard Zumkeller, Aug 07 2011

nmax:=70: 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) := 2^(p+1)*n-1 od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Feb 01 2013

Table[BitOr[(n + 1), n], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 19 2011 *)

a(n)=bitor(n,n-1) \\ Charles R Greathouse IV, Apr 17 2012

def a(n): return n | (n-1)
print([a(n) for n in range(1, 71)]) # Michael S. Branicky, Jul 13 2022

a(n) = n + 2^A007814(n) - 1.
a(n) is odd; a(n) = n iff n is odd.
a(a(n)) = a(n); A007814(a(n)) = a(n); A000265(a(n)) = a(n).
A023416(a(n)) = A023416(n) - A007814(n) = A086784(n).
A000120(a(n)) = A000120(n) + A007814(n).
a(2^n) = a(A000079(n)) = 2*2^n - 1 = A000051(n+1).
a(n) = if n is odd then n else a(n/2)*2 + 1.
a(n) = A006519(n) + n - 1. - Reinhard Zumkeller, Feb 02 2007
a(n) = n OR n-1 (bitwise OR of consecutive numbers). - Russ Cox, May 15 2007
a(2*n) = A038712(n) + 2*n. - Reinhard Zumkeller, Aug 07 2011
a((2*n-1)*2^p) = 2^(p+1)*n-1, p >= 0. - Johannes W. Meijer, Feb 01 2013
Sum_{k=1..n} a(k) ~ n^2/2 + (1/(2*log(2)))*n*log(n) + (3/4 + (gamma-1)/(2*log(2)))*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 24 2022

A090739 Exponent of 2 in 9^n - 1.

Original entry on oeis.org

3, 4, 3, 5, 3, 4, 3, 6, 3, 4, 3, 5, 3, 4, 3, 7, 3, 4, 3, 5, 3, 4, 3, 6, 3, 4, 3, 5, 3, 4, 3, 8, 3, 4, 3, 5, 3, 4, 3, 6, 3, 4, 3, 5, 3, 4, 3, 7, 3, 4, 3, 5, 3, 4, 3, 6, 3, 4, 3, 5, 3, 4, 3, 9, 3, 4, 3, 5, 3, 4, 3, 6, 3, 4, 3, 5, 3, 4, 3, 7, 3, 4, 3, 5, 3, 4, 3

Offset: 1

Labos Elemer and Ralf Stephan, Jan 19 2004

The exponent of 2 in the factorization of Fibonacci(6n). - T. D. Noe, Mar 14 2014

Records of 3, 4, 5, 6, 7, 8,.. occur at n= 1, 2, 4, 8, 16, 32,... - R. J. Mathar, Jun 28 2025

			For n = 2, we see that -1 + 3^4 = 80 = 2^4 * 5 so a(2) = 4.
For n = 3, we see that -1 + 3^6 = 728 = 2^3 * 7 * 13, so a(3) = 3.

T. D. Noe, Table of n, a(n) for n = 1..1000
T. Lengyel, The order of the Fibonacci and Lucas numbers, Fib. Quart. 33 (1995), 234-239.

Cf. A024101, A069895, A091512, A088660, A090740, A220466.
Cf. A000005, A006519, A120738 (partial sums).
Appears in A161737.

A090739 := proc(n)
    padic[ordp](9^n-1,2) ;
end proc:
seq(A090739(n),n=1..80) ; # R. J. Mathar, Jun 28 2025

Table[Part[Flatten[FactorInteger[ -1+3^(2*n)]], 2], {n, 1, 70}]
Table[IntegerExponent[Fibonacci[n], 2], {n, 6, 600, 6}] (* T. D. Noe, Mar 14 2014 *)

a(n)=valuation(n,2)+3 \\ Charles R Greathouse IV, Mar 14 2014

def A090739(n): return (~n&n-1).bit_length()+3 # Chai Wah Wu, Jul 11 2022

a(n) = A007814(n) + 3.
a((2*n-1)*2^p) = p + 3, p >= 0. - Johannes W. Meijer, Feb 08 2013
a(n) = log_2(A006519(9^n - 1)). - Alonso del Arte, Feb 08 2013
a(n) = 2*tau(4*n)/(tau(4*n) - tau(n)), where tau(n) = A000005(n). - Peter Bala, Jan 06 2021
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4. - Amiram Eldar, Nov 28 2022

More terms from T. D. Noe, Mar 14 2014

A091512 a(n) is the largest integer m such that 2^m divides (2n)^n, i.e., the exponent of 2 in (2n)^n.

Original entry on oeis.org

1, 4, 3, 12, 5, 12, 7, 32, 9, 20, 11, 36, 13, 28, 15, 80, 17, 36, 19, 60, 21, 44, 23, 96, 25, 52, 27, 84, 29, 60, 31, 192, 33, 68, 35, 108, 37, 76, 39, 160, 41, 84, 43, 132, 45, 92, 47, 240, 49, 100, 51, 156, 53, 108, 55, 224, 57, 116, 59, 180, 61, 124, 63

Offset: 1

Ralf Stephan and Labos Elemer, Jan 18 2004

n times one more than the trailing 0's in the binary representation of n. - Ralf Stephan, Aug 22 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n.

Cf. A091519, A090740, A090739, A048298, A162728, A220466.

[n*(Valuation(n, 2)+1): n in [1..80]]; // Vincenzo Librandi, May 16 2013

nmax:=63: 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) := (2*n-1)*(p+1)*2^p od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Feb 08 2013
# second Maple program:
a:= n-> n*padic[ordp](2*n, 2):
seq(a(n), n=1..63);  # Alois P. Heinz, Jan 02 2024

Table[ Part[ Flatten[ FactorInteger[(2 n)^n]], 2], {n, 1, 124}]
Table[IntegerExponent[(2n)^n,2],{n,70}] (* Harvey P. Dale, Sep 11 2015 *)

PARI
```
a(n)=n*(valuation(n,2)+1)
```
PARI
```
a(n)=if(n<1,0,if(n%2==0,2*a(n/2)+n,n))
```

def A091512(n): return n*(n&-n).bit_length() # Chai Wah Wu, Jul 11 2022

a(n) = A007814(A000312(n)) = n*A001511(n) = A069895(n)/2.
G.f.: Sum_{k>=0} 2^k*x^2^k/(1-x^2^k)^2.
Recurrence: a(0) = 0, a(2*n) = 2*a(n) + 2*n, a(2*n+1) = 2*n+1.
Dirichlet g.f.: zeta(s-1)*2^s/(2^s-2). - Ralf Stephan, Jun 17 2007
Mobius transform of A162728, where x/(1-x)^2 = Sum_{n>=1} A162728(n)*x^n/(1+x^n). - Paul D. Hanna, Jul 12 2009
a(n) = A162728(2*n)/phi(2*n), where x/(1-x)^2 = Sum_{n>=1} A162728(n)*x^n/(1+x^n). - Paul D. Hanna, Jul 12 2009
a((2*n-1)*2^p) = (2*n-1)*(p+1)*2^p, p >= 0. Observe that a(2^p) = A001787(p+1). - Johannes W. Meijer, Feb 08 2013
Sum_{k=1..n} a(k) ~ n^2. - Amiram Eldar, Oct 22 2022
a(n) = Sum_{d divides n} d*A048298(n/d); that is, a(n) is the Dirichlet product of A048298(n) and A000027(n). - Peter Bala, Jan 02 2024

cp's OEIS Frontend

A048896 a(n) = 2^(A000120(n+1) - 1), n >= 0.

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A096268 Period-doubling sequence (or period-doubling word): fixed point of the morphism 0 -> 01, 1 -> 00.

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A060789 a(n) = n / (gcd(n,2) * gcd(n,3)).

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A129760 Bitwise AND of binary representation of n-1 and n.

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A014707 a(4n) = 0, a(4n+2) = 1, a(2n+1) = a(n).

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A037227 If n = 2^mk, k odd, then a(n) = 2m+1.