

A007733


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


54



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
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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
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, 7188, 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


FORMULA

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


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
 Reinhard Zumkeller, Apr 13 2015
(Python)
from sympy.ntheory import n_order
def A007733(n): return n_order(2, n>>(~n & n1).bit_length()) # Chai Wah Wu, Jul 01 2022


CROSSREFS

Cf. A136042.  John W. Layman, Jan 22 2009
Cf. A000265, A002326, A256607, A256757.
Sequence in context: A183200 A326732 A305422 * A128520 A269370 A123755
Adjacent sequences: A007730 A007731 A007732 * A007734 A007735 A007736


KEYWORD

nonn,easy


AUTHOR

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


STATUS

approved



