

A007733


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


56



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


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 base2 MRexpansion of 1/n (see A136042 for definition).  John W. Layman, Jan 22 2009
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, 7188, see Table 2. Math. Rev. 95f:05052.


LINKS



FORMULA



MATHEMATICA

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


PROG

(PARI) a(n) = znorder(Mod(2, n/2^valuation(n, 2))); \\ Michel Marcus, Apr 11 2015
(Haskell)
a007733 = a002326 . flip div 2 . subtract 1 . a000265
(Python)
from sympy.ntheory import n_order


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



