A091317 -id:A091317 - cp's OEIS frontend

A014663 Primes p such that multiplicative order of 2 modulo p is odd.

7, 23, 31, 47, 71, 73, 79, 89, 103, 127, 151, 167, 191, 199, 223, 233, 239, 263, 271, 311, 337, 359, 367, 383, 431, 439, 463, 479, 487, 503, 599, 601, 607, 631, 647, 719, 727, 743, 751, 823, 839, 863, 881, 887, 911, 919, 937, 967, 983, 991, 1031, 1039, 1063

Offset: 1

N. J. A. Sloane, Dec 11 1999

nonn

Or, primes p which do not divide 2^n+1 for any n.
The possibility n=0 in the above rules out A072936(1)=2; apart from this, a(n)=A072936(n+1). - M. F. Hasler, Dec 08 2007
The order of 2 mod p is odd iff 2^k=1 mod p, where p-1=2^s*k, k odd. - M. F. Hasler, Dec 08 2007
Has density 7/24 (Hasse).
From Jianing Song, Jun 27 2025: (Start)
The multiplicative order of 2 modulo a(n) is A139686(n).
Contained in primes congruent to 1 or 7 modulo 8 (primes p such that 2 is a quadratic residue modulo p, A001132), and contains primes congruent to 7 modulo 8 (A007522). (End)

Christopher Adler and Jean-Paul Allouche (2022), Finite self-similar sequences, permutation cycles, and music composition, Journal of Mathematics and the Arts, 16:3, 244-261, DOI: 10.1080/17513472.2022.2116745.
P. Moree, Appendix to V. Pless et al., Cyclic Self-Dual Z_4 Codes, Finite Fields Applic., vol. 3 pp. 48-69, 1997.

T. D. Noe, Table of n, a(n) for n=1..1000
H. H. Hasse, Über die Dichte der Primzahlen p, ... , Math. Ann., 168 (1966), 19-23.
J. C. Lagarias, The set of primes dividing the Lucas numbers has density 2/3, Pacific J. Math., 118. No. 2, (1985), 449-461.
Chunlei Li, Nian Li, and Matthew G. Parker, Complementary Sequence Pairs of Types II and III. [From _N. J. A. Sloane_, Jun 16 2012]

Cf. Complement in primes of A091317.
Cf. A001132, A007522, A040098, A045315, A049564, A139686 (the actual multiplicative orders).
Cf. Essentially the same as A072936 (except for missing leading term 2).
Cf. other bases: this sequence (base 2), A385220 (base 3), A385221 (base 4), A385192 (base 5), A163183 (base -2), A385223 (base -3), A385224 (base -4), A385225 (base -5).

okQ[p_] := OddQ[MultiplicativeOrder[2, p]];
Select[Prime[Range[1000]], okQ] (* Jean-François Alcover, Nov 23 2024 *)

isA014663(p)=1==Mod(1,p)<<((p-1)>>factor(p-1,2)[1,2])
listA014663(N=1000)=forprime(p=3,N,isA014663(p)&print1(p", ")) \\ M. F. Hasler, Dec 08 2007

lista(nn) = {forprime(p=3, nn, if (znorder(Mod(2, p)) % 2, print1(p, ", ")););} \\ Michel Marcus, Feb 06 2015

Edited by M. F. Hasler, Dec 08 2007

More terms from Max Alekseyev, Feb 06 2010

A014657 Numbers m that divide 2^k + 1 for some nonnegative k.

Original entry on oeis.org

1, 2, 3, 5, 9, 11, 13, 17, 19, 25, 27, 29, 33, 37, 41, 43, 53, 57, 59, 61, 65, 67, 81, 83, 97, 99, 101, 107, 109, 113, 121, 125, 129, 131, 137, 139, 145, 149, 157, 163, 169, 171, 173, 177, 179, 181, 185, 193, 197, 201, 205, 209, 211, 227, 229, 241, 243, 249, 251, 257, 265

Offset: 1

N. J. A. Sloane

Since for some a < n, 2^a == 1 (mod n) (a consequence of Euler's Theorem), searching up to k=n is sufficient to determine whether an integer is in the sequence. - Michael B. Porter, Dec 06 2009
A195470(a(n)) > 0; A195610(n) gives the smallest k such that a(n) divides 2^k + 1. - Reinhard Zumkeller, Sep 21 2011
This sequence is the subset of odd integers > 1 as (2*n - 1) in A179480, such that the corresponding entry in A179480 is odd. Example: A179480(14) = 5, odd, with (2*14 - 1) = 27; and 5 is a term of this sequence. A014659 (odd and does not divide (2^k + 1) for any k >= 1) represents the subset of odd terms >1 corresponding to A179480 entries that are even. - Gary W. Adamson, Aug 20 2012
All prime factors of a(n) are in A091317.  Sequence has asymptotic density 0. - Robert Israel, Aug 12 2014
This sequence, for m>2, is those m for which, for some e, (m-1)(2^e-1)/m is a term of A253608. Moreover, e(n) is 2*A195610(n) when m is a(n). - Donald M Davis, Jan 12 2018
From Wolfdieter Lang, Aug 22 2020: (Start)
Without a(2) = 2 this is the complement of A014659 relative to the odd positive integers A005408.
For the least nonnegative integer k(n) with 2^k(n) + 1 == d(n)*a(n), for n >= 1, see k(n) = A195610(n) and d(n) = A337220(n).
Starting with a(3) = 3 these numbers are the odd moduli, named 2*n+1 in the definition of A003558, for which the minus signs applies (see A332433(m) for the signs applying for A003558(m)). (End)

T. D. Noe, Table of n, a(n) for n = 1..1000
P. Moree, Appendix to V. Pless et al., Cyclic Self-Dual Z_4 Codes, Finite Fields Applic., vol. 3 pp. 48-69, 1997.

Besides initial terms 1 and 2, a subsequence of A296243. Their set difference is given by A296244.

Cf. A000051, A003558, A005408, A014659, A014661, A091317, A179480, A195470, A195610, A332433, A337220.

import Data.List (findIndices)
a014657 n = a014657_list !! (n-1)
a014657_list = map (+ 1) $ findIndices (> 0) $ map a195470 [1..]
-- Reinhard Zumkeller, Sep 21 2011

select(t -> [msolve(2^x+1,t)] <> [], [2*i+1 $ i=1..1000]); # Robert Israel, Aug 12 2014

ok[n_] := Module[{k=0}, While[k<=n && Mod[2^k + 1, n] > 0, k++]; kJean-François Alcover, Apr 06 2011, after PARI prog *)
okQ[n_] := Module[{k = MultiplicativeOrder[2, n]}, EvenQ[k] && Mod[2^(k/2) + 1, n] == 0]; Join[{1, 2}, Select[Range[3, 265, 2], okQ]] (* T. D. Noe, Apr 06 2011 *)

isA014657(n) = {local(r);r=0;for(k=0,n,if(Mod(2^k+1,n)==Mod(0,n),r=1));r} \\ Michael B. Porter, Dec 06 2009

More terms from Henry Bottomley, May 19 2000

Extended and corrected by David W. Wilson, May 01 2001

A014662 Primes p such that order of 2 mod p (=A007733(p)) is even.

Original entry on oeis.org

3, 5, 11, 13, 17, 19, 29, 37, 41, 43, 53, 59, 61, 67, 83, 97, 101, 107, 109, 113, 131, 137, 139, 149, 157, 163, 173, 179, 181, 193, 197, 211, 227, 229, 241, 251, 257, 269, 277, 281, 283, 293, 307, 313, 317, 331, 347, 349, 353, 373, 379, 389, 397, 401, 409, 419

Offset: 1

N. J. A. Sloane

nonn

Apart from the first term, identical to A091317. - Charles R Greathouse IV, Feb 13 2009

Dirichlet density is 5/24 (Fein, Gordon, & Smith); they show a result on expressing -1 as the sum of two squares relating to this sequence. - Charles R Greathouse IV, May 15 2024

P. Moree, Appendix to V. Pless et al., Cyclic Self-Dual Z_4 Codes, Finite Fields Applic., vol. 3 pp. 48-69, 1997.

Klaus Brockhaus, Table of n, a(n) for n = 1..1000
Burton Fein, Basil Gordon, and John H. Smith, On the representation of -1 as a sum of two squares in an algebraic number field J. Num. Theor. (1971) Vol. 3, Issue 3, 310-315.
Eugen J. Ionascu, Florian Luca, and Thomas Merino, On the average value of the minimal Hamming multiple, arXiv:2412.10839 [math.NT], 2024. See pp. 4, 17.
C. Smyth, The terms in Lucas Sequences divisible by their indices, JIS 13 (2010) #10.2.4.

The prime terms of A296243.

Cf. A091317.

[ p: p in PrimesInInterval(3, 419) | IsEven(Modorder(2, p)) ]; // Klaus Brockhaus, Dec 09 2008

select(t -> isprime(t) and numtheory:-order(2,t)::even, [2*i+1 $ i=1..1000]); # Robert Israel, Aug 12 2014

Select[Prime[Range[80]], EvenQ[MultiplicativeOrder[2, #/(2^IntegerExponent[ #, 2])]]&] (* Jean-François Alcover, Sep 02 2018 *)

isok(p) = isprime(p) && !(znorder(Mod(2, p/2^valuation(p, 2))) % 2); \\ Michel Marcus, Sep 02 2018

is(n)=n>2 && Mod(2,n)^(n>>valuation(n-1,2))!=1 && isprime(n) \\ Charles R Greathouse IV, May 07 2024

More terms from Klaus Brockhaus, Dec 09 2008

cp's OEIS Frontend

A014663 Primes p such that multiplicative order of 2 modulo p is odd.

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A014657 Numbers m that divide 2^k + 1 for some nonnegative k.

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A014662 Primes p such that order of 2 mod p (=A007733(p)) is even.

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Author

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References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

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Extensions