1,2

Least positive k such that m^k+1 divides m^n+1(with fixed base m). - Vladimir Baltic (baltic(AT)matf.bg.ac.yu), Mar 25 2002

To construct the sequence: start with 1, concatenate 1,1 and double last term gives 1,2. Concatenate those 2 terms, 1,2,1,2 and double last term 1,2,1,2 ->1,2,1,4. Concatenate those 4 terms: 1,2,1,4,1,2,1,4 and double last term -> 1,2,1,4,1,2,1,8 etc. - Benoit Cloitre (benoit7848c(AT)orange.fr), Dec 17 2002

a(n)=GCD(seq(binomial(2*n,2*m+1)/2,m=0..n-1)) (odd numbered entries of even numbered rows of Pascal's triangle A007318 divided by 2), where GCD denotes the greatest common divisor of a set of numbers. Due to the symmetry of the rows it suffices to consider m=0..floor((n-1)/2). Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Jan 23 2004

Equals the continued fraction expansion of a constant x (cf. A100338) such that the continued fraction expansion of 2*x interleaves this sequence with 2's: contfrac(2*x) = [2; 1,2, 2,2, 1,2, 4,2, 1,2, 2,2, 1,2, 8,2,...].

Simon Plouffe (simon.plouffe(AT)gmail.com) observes that this sequence and A003484 (Radon function) are very similar, the difference being all zeros except for every 16-th term (see A101119 for nonzero differences). Dec 02, 2004.

Comment from Jim Caprioli, Feb 04 2005: This sequence arises when calculating the next odd number in a Collatz sequence: Next(x) = (3 * x + 1) / A006519, or simply (3 x + 1) / BitAnd(3x+1,-3x-1).

a(n) = n iff n = 2^k. This sequence can be obtained by taking a(2^n) = 2^n inplace of a(2^n) = n and using the same sequence building approach as in A001511. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jul 08 2005

Also smallest m such that m + n - 1 = m XOR (n - 1); A086799(n)=a(n)+n-1. - Reinhard Zumkeller, Feb 02 2007

Number of 1's between successive 0's in A159689. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 22 2009]

Least number k such that all coefficients of k*E(n,x), the n-th Euler polynomial, are integers (Cf. A144845). [From Peter Luschny (peter(AT)luschny.de), Nov 13 2009]

R. Brown and J. L. Merzel, The number of Ducci sequences with a given period, Fib. Quart., 45 (2007), 115-121.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..10000

Beeler, M., Gosper, R. W. and Schroeppel, R., Item 175 in Beeler, M., Gosper, R. W. and Schroeppel, R. HAKMEM. MIT AI Memo 239, Feb. 29, 1972

R. Stephan, Some divide-and-conquer sequences ...

R. Stephan, Table of generating functions

R. Stephan, Divide-and-conquer generating functions. I. Elementary sequences

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

a(n) = n AND -n (where "AND" is bitwise) - Marc LeBrun (mlb(AT)well.com), Sep 25 2000

Also: a(n)=gcd[2^n, n]. - Labos E. (labos(AT)ana.sote.hu), Apr 22 2003

Multiplicative with a(p^e) = p^e if p = 2; 1 if p > 2. - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.

G.f.: sum(k>=0, 2^k*x^2^k/(1-x^2^(k+1))). - Ralf Stephan, May 06 2003

Dirichlet g.f.: zeta(s)*(2^s-1)/(2^s-2). - Ralf Stephan, Jun 17 2007

a(n) = 2^floor(A002487(n - 1) / A002487(n)). [From Reikku Kulon (reikku(AT)gmail.com), Oct 05 2008]

2^3 divides 24, but 2^4 does not divide 24, so a(24)=8.

with(numtheory): for n from 1 to 200 do if n mod 2 = 1 then printf(`%d, `, 1) else printf(`%d, `, 2^ifactors(n)[2][1][2]) fi; od:

f[n_] := Block[{k = 0}, While[Mod[n, 2^k] == 0, k++ ]; 2^(k - 1)]; Table[ f[n], {n, 102}] (from Robert G. Wilson v Nov 17 2004)

(PARI) a(n)=2^valuation(n, 2)

Partial sums are in A006520, second partial sums in A022560.

Cf. A007814, A100338, A003484, A101119.

This is Guy Steele's sequence GS(5, 2) (see A135416).

Cf. A002487 [From Reikku Kulon (reikku(AT)gmail.com), Oct 05 2008]

Sequence in context: A084236 A068057 A003484 this_sequence A055975 A118827 A118830

Adjacent sequences: A006516 A006517 A006518 this_sequence A006520 A006521 A006522

nonn,easy,nice,mult,new

N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jun 20 2000