

A171977


a(n) = 2^(k+1) where 2^k is the highest power of 2 dividing n.


11



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
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OFFSET

1,1


COMMENTS

When read as a triangle in which the nth 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,2n1)].  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
After the all1's sequence this is the next sequence in lexicographical order such that the gap between a(n) and the next occurrence of a(n) is given by a(n).  Scott R. Shannon, Oct 16 2019
First 2^(k1)  1 terms are also the areas of the successive rectangles and squares of width 2 that are adjacent to any of the four sides of the toothpick structure of A139250 after 2^k stages, with k >= 2. For example: if k = 5 the areas after 32 stages are [2, 4, 2, 8, 2, 4, 2, 16, 2, 4, 2, 8, 2, 4, 2] respectively, the same as the first 15 terms of this sequence.  Omar E. Pol, Dec 29 2020


LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16383
Sandor Csörgö and Gordon Simons, On Steinhaus' resolution of the St. Petersburg paradox, Probab. Math. Statist. 14 (1993), 157172. MR1321758 (96b:60017).  Peter Kern, Aug 28 2015
Roger B. Eggleton, Aviezri S. Fraenkel, and R. Jamie Simpson, Beatty sequences and Langford sequences, Graph theory and combinatorics (MarseilleLuminy, 1990). Discrete Math. 111 (1993), no. 13, 165178. MR1210094 (94a:11018). See Example 2.6.  N. J. A. Sloane, Mar 18 2012
Hugo Steinhaus, The socalled Petersburg paradox, Colloq. Math. 2 (1949), 5658. 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(n1).  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 := k1: a(n) := 2^(k+1) end: seq(a(n), n=1..63); # Johannes W. Meijer, Nov 04 2012
seq(2^(1 + padic[ordp](n, 2)), n = 1..63); # Peter Luschny, Nov 27 2020


MATHEMATICA

Table[BitXor[i, i], {i, 200}] (* Peter Luschny, Jun 01 2011 *)
a[n_] := 2^(IntegerExponent[n, 2] + 1); Array[a, 100] (* JeanFrançois Alcover, May 09 2017 *)


PROG

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


CROSSREFS

Cf. A000079, A000265, A006519, A038712, A139250.
Sequence in context: A259111 A209675 A307669 * 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



