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A171977 a(n) = 2^(k+1) where 2^k is the highest power of 2 dividing n. 8
2, 4, 2, 8, 2, 4, 2, 16, 2, 4, 2, 8, 2, 4, 2, 32, 2, 4, 2, 8, 2, 4, 2, 16, 2, 4, 2, 8, 2, 4, 2, 64, 2, 4, 2, 8, 2, 4, 2, 16, 2, 4, 2, 8, 2, 4, 2, 32, 2, 4, 2, 8, 2, 4, 2, 16, 2, 4, 2, 8, 2, 4, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

When read as a triangle in which the n-th row has 2^n terms, every row is the last half of the next one. All the terms are powers of 2. First column = 2*A000079.

The original definition was: a(n) = (A000265(2n+1) - 1) / A000265(2n).

a(n) seems to be the denominator of Euler(2*n+1,1) but I have no proof of this.

a(n) is also gcd[C(2n,1), C(2n,3), ..., C(2n,2n-1)]. - Franz Vrabec, Oct 22 2012

a(n) is also the ratio r(2n) = s2(2n)/s1(2n) where s1(2n) is the sum of the odd unitary divisors of 2n and s2(2n) is the sum of the even unitary divisors of 2n. - Michel Lagneau, Dec 19 2013

a(n) or a(n)/2 = A006519(n) is known as the Steinhaus sequence in probability theory, proposed as a sequence of asymptotically fair premiums for the St. Petersburg game. - Peter Kern, Aug 28 2015

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16383

Sandor Csörgö, Gordon Simons, On Steinhaus' resolution of the St. Petersburg paradox, Probab. Math. Statist. 14 (1993), 157--172. MR1321758 (96b:60017). - Peter Kern, Aug 28 2015

Roger B. Eggleton, Aviezri S. Fraenkel, R. Jamie Simpson, Beatty sequences and Langford sequences, Graph theory and combinatorics (Marseille-Luminy, 1990). Discrete Math. 111 (1993), no. 1-3, 165--178. MR1210094 (94a:11018). See Example 2.6. - N. J. A. Sloane, Mar 18 2012

Hugo Steinhaus, The so-called Petersburg paradox, Colloq. Math. 2 (1949), 56--58. MR0039937 (12,619e).

FORMULA

a(n) = (A000265(2*n+1)-1)/A000265(2*n).

a(n) = -(-n XOR n).  XOR the bitwise operation on the two's complement representation for negative integers. - Peter Luschny, Jun 01 2011

a(n) = A038712(n)+1. - Franz Vrabec, Mar 03 2012

a(n) = 2^A001511(n). - Franz Vrabec, Oct 22 2012

a(n) = A046161(n)/A046161(n-1). - Johannes W. Meijer, Nov 04 2012

a(n) = 2^(1 + (A183063(n)/A001227(n))). - Omar E. Pol, Nov 06 2018

a(n) = 2*A006519(n). - Antti Karttunen, Nov 06 2018

MAPLE

a := proc(n) local k: k:=1: while frac(n/2^k) = 0 do k := k+1 end do: k := k-1: a(n) := 2^(k+1) end: seq(a(n), n=1..63); # Johannes W. Meijer, Nov 04 2012

MATHEMATICA

Table[-BitXor[-i, i], {i, 200}] (* Peter Luschny, Jun 01 2011 *)

a[n_] := 2^(IntegerExponent[n, 2] + 1); Array[a, 100] (* Jean-François Alcover, May 09 2017 *)

PROG

(PARI) A171977(n) = 2^(1+valuation(n, 2)); \\ Antti Karttunen, Nov 06 2018

CROSSREFS

Cf. A000079, A000265, A006519, A038712, A171977.

Sequence in context: A296092 A259111 A209675 * A266073 A059866 A278262

Adjacent sequences:  A171974 A171975 A171976 * A171978 A171979 A171980

KEYWORD

nonn,tabf

AUTHOR

Paul Curtz, Nov 19 2010

EXTENSIONS

I edited this sequence, based on an email message from the author. - N. J. A. Sloane, Nov 20 2010

Definition simplified by N. J. A. Sloane, Mar 18 2012

STATUS

approved

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Last modified January 20 12:32 EST 2019. Contains 319330 sequences. (Running on oeis4.)