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

1,1

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).

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

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]

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. A040098, A045315, A049564.

Cf. Essentially the same as A072936 (except for missing leading term 2).

Sequence in context: A004628 A089199 A263874 * A007522 A141175 A295196

Adjacent sequences: A014660 A014661 A014662 * A014664 A014665 A014666

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