A139099 -id:A139099 - cp's OEIS frontend

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

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

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

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.

Cf. A136042. - John W. Layman, Jan 22 2009
Cf. A000265, A002326, A256607, A256757.
Positions of records are A139099.

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

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

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

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

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

A179383 a(n) = 2*k(n)-1 where k(n) is the sequence of positions of records in A179382.

Original entry on oeis.org

1, 5, 9, 11, 13, 19, 25, 29, 37, 53, 59, 61, 67, 83, 101, 107, 121, 131, 139, 149, 163, 173, 179, 181, 197, 211, 227, 269, 293, 317, 347, 349, 373, 379, 389, 419, 421, 443, 461, 467, 491, 509, 523, 541, 547, 557, 563, 587, 613, 619, 653, 659, 661, 677, 701, 709, 757

Offset: 1

Vladimir Shevelev, Jul 12 2010

nonn

Records in A179382(k(n)) = 1, 2, 3, 5, 6, 9, 10, 14, 18, 26, 29, ....
are located at k(n) = 1, 3, 5, 6, 7, 10, 13, 15, 19, 27, 30, 31,..
The current sequence is a simple transformation of this k(n) sequence.
Question: Are there any terms in the sequence with two or more distinct prime divisors?
Some very plausible conjectures: 1) The sequence consists of primes and squares of primes; 2) The set of squares is finite; 3) A prime p>=5 is in the sequence iff it has primitive root 2 (A001122) ; 4) There exists l such that, for n>l, A179383(n) =A139099(n+l) . [From Vladimir Shevelev , Jul 14 2010]

Peter J. C. Moses, Table of n, a(n) for n = 1..8500

Cf. A139099, A167791, A002326, A179382

Definition rephrased and sequence extended by R. J. Mathar, Jul 13 2010

I made a change to Conjecture 4). - Vladimir Shevelev, Jul 18 2010

A179538 Numbers 2n+1 for which A003558(n), n>=1, are record values of A003558.

Original entry on oeis.org

3, 5, 7, 11, 13, 19, 23, 29, 37, 47, 53, 59, 61, 67, 71, 79, 83, 101, 103, 107, 121, 131, 139, 149, 163, 167, 173, 179, 181, 191, 197, 199, 211, 227, 239, 263, 269, 271, 293, 311, 317, 347, 349, 359, 367, 373, 379, 383, 389, 419, 421, 443, 461, 463, 467, 479

Offset: 1

Vladimir Shevelev, Jul 18 2010

nonn

Conjecture. All terms are primes except for a finite set of squares of primes.

All terms from a(1) to a(5000) are primes except for a(21) = 121 = 11^2, supporting V. Shevelev's conjecture. [John W. Layman, Jul 22 2010]

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A003558, A139099, A179383, A179481.

s = {}; am = 0; Do[a = Min[MultiplicativeOrder[2, n, {-1, 1}]]; If[a > am, am = a; AppendTo[s, n]], {n, 3, 480, 2}]; s (* Amiram Eldar, Sep 13 2019 *)

a(31)-a(56) from John W. Layman, Jul 22 2010

cp's OEIS Frontend

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

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A179383 a(n) = 2*k(n)-1 where k(n) is the sequence of positions of records in A179382.

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A179538 Numbers 2n+1 for which A003558(n), n>=1, are record values of A003558.

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Mathematica

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