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%I #58 Dec 21 2022 12:00:39
%S 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,
%T 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,
%U 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
%N Period of binary representation of 1/n. Also, multiplicative order of 2 modulo the odd part of n (= A000265(n)).
%C 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
%C 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
%C 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
%C Indices n such that a(n) divides n are listed in A068563. - _Max Alekseyev_, Aug 25 2013
%C 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
%D 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.
%H T. D. Noe, <a href="/A007733/b007733.txt">Table of n, a(n) for n=1..1000</a>
%H <a href="/index/1#1overn">Index entries for sequences related to decimal expansion of 1/n</a>
%F a(n) = A002326((A000265(n) - 1)/2). - _Max Alekseyev_, Jun 11 2009
%t f[n_] := MultiplicativeOrder[2, n/(2^IntegerExponent[n, 2])]; Array[f, 84] (* _Robert G. Wilson v_, Jun 10 2011 *)
%o (PARI) a(n) = znorder(Mod(2, n/2^valuation(n, 2))); \\ _Michel Marcus_, Apr 11 2015
%o (Haskell)
%o a007733 = a002326 . flip div 2 . subtract 1 . a000265
%o -- _Reinhard Zumkeller_, Apr 13 2015
%o (Python)
%o from sympy.ntheory import n_order
%o def A007733(n): return n_order(2,n>>(~n & n-1).bit_length()) # _Chai Wah Wu_, Jul 01 2022
%Y Cf. A136042. - _John W. Layman_, Jan 22 2009
%Y Cf. A000265, A002326, A256607, A256757.
%Y Positions of records are A139099.
%K nonn,easy
%O 1,3
%A _N. J. A. Sloane_, Hal Sampson (hals(AT)easynet.com)
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