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URL: https://oeis.org/A014663

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A014663
Primes p such that multiplicative order of 2 modulo p is odd.
21
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
1,1
COMMENTS
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)
REFERENCES
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.
LINKS
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]
MATHEMATICA
okQ[p_] := OddQ[MultiplicativeOrder[2, p]];
Select[Prime[Range[1000]], okQ] (* Jean-François Alcover, Nov 23 2024 *)
PROG
(PARI) 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
(PARI) lista(nn) = {forprime(p=3, nn, if (znorder(Mod(2, p)) % 2, print1(p, ", ")); ); } \\ Michel Marcus, Feb 06 2015
CROSSREFS
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).
Sequence in context: A004628 A089199 A263874 * A007522 A141175 A295196
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Dec 11 1999
EXTENSIONS
Edited by M. F. Hasler, Dec 08 2007
More terms from Max Alekseyev, Feb 06 2010
STATUS
approved