A007733 Period of binary representation of 1/n. Also, multiplicative order of 2 modulo the odd part of n (= A000265(n)).

1, 1, 2, 1, 4, 2, 3, 1, 6, 4, 10, 2, 12, 3, 4, 1, 8, 6, 18, 4, 6, 10, 11, 2, 20, 12, 18, 3, 28, 4, 5, 1, 10, 8, 12, 6, 36, 18, 12, 4, 20, 6, 14, 10, 12, 11, 23, 2, 21, 20, 8, 12, 52, 18, 20, 3, 18, 28, 58, 4, 60, 5, 6, 1, 12, 10, 66, 8, 22, 12, 35, 6, 9, 36, 20, 18, 30, 12, 39, 4, 54, 20, 82, 6

Offset: 1

Author

N. J. A. Sloane, Hal Sampson (hals(AT)easynet.com)

Keywords

• nonn
• easy

Comments

Also sequence of period lengths for n's when you do primality testing and calculate "2^k mod n" from k = 0..n. - Gottfried Helms, Oct 05 2000

Fractal sequence related to A002326: the even terms of this sequence are this sequence itself, constructed on A002326, whose terms are the odd terms of this sequence. - Alexandre Wajnberg, Apr 27 2005

It seems that a(n) is also the sum of the terms in one period of the base-2 MR-expansion of 1/n (see A136042 for definition). - John W. Layman, Jan 22 2009

Indices n such that a(n) divides n are listed in A068563. - Max Alekseyev, Aug 25 2013

a(n) is the smallest k such that x^n - 1 factors into n linear polynomials over GF(2^k). For example, a(12) = 2, and x^12 - 1 = (x - 1)^4*(x - w)^4*(x - (w + 1))^4 in GF(4), where w^2 + w + 1 = 0. - Jianing Song, Jan 20 2019

References

• Simmons, G. J. The structure of the differentiation digraphs of binary sequences. Ars Combin. 35 (1993), A, 71-88, see Table 2. Math. Rev. 95f:05052.

Links

• T. D. Noe, Table of n, a(n) for n=1..1000
• Index entries for sequences related to decimal expansion of 1/n

Crossrefs

Cf. A136042. - John W. Layman, Jan 22 2009

Cf. A000265, A002326, A256607, A256757.

Positions of records are A139099.

Programs

Haskell

a007733 = a002326 . flip div 2 . subtract 1 . a000265
-- Reinhard Zumkeller, Apr 13 2015

Mathematica

f[n_] := MultiplicativeOrder[2, n/(2^IntegerExponent[n, 2])]; Array[f, 84] (* Robert G. Wilson v, Jun 10 2011 *)

PARI

a(n) = znorder(Mod(2, n/2^valuation(n, 2))); \\ Michel Marcus, Apr 11 2015

Python

from sympy.ntheory import n_order
def A007733(n): return n_order(2,n>>(~n & n-1).bit_length()) # Chai Wah Wu, Jul 01 2022

Formula

a(n) = A002326((A000265(n) - 1)/2). - Max Alekseyev, Jun 11 2009