A049564 -id:A049564

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A014663

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

+10
9

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

(list; graph; refs; listen; history; text; internal format)

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]

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. A040098, A045315, A049564.

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

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

A072936

Primes p that do not divide 2^x+1 for any x>=1.

+10
6

2, 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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Also, primes p such that p^2 does not divide 2^x+1 for any x>=1.

A prime p cannot divide 2^x+1 if the multiplicative order of 2 (mod p) is odd. - T. D. Noe, Aug 22 2004

Differs from A049564 first at p=6529, which is the 250th entry in A049564 related to 279^32 =2 mod 6529, but absent here because 6529 divides 2^51+1. [From R. J. Mathar, Sep 25 2008]

REFERENCES

A. K. Devaraj, "Euler's Generalization of Fermat's Theorem-A Further Generalization", in ISSN #1550-3747, Proceedings of Hawaii Intl Conference on Statistics, Mathematics & Related Fields, 2004.

LINKS

T. D. Noe, Table of n, a(n) for n=1..1000

CROSSREFS

Cf. A040098, A049096, A014664 (multiplicative order of 2 mod n-th prime).

KEYWORD

nonn

AUTHOR

Benoit Cloitre, Aug 20 2002

EXTENSIONS

Edited by T. D. Noe, Aug 22 2004

STATUS

approved

A059349

Primes p such that x^32 = 2 has no solution mod p.

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Complement of A049564 relative to A000040.

Differs from A014662 first at p=6529, then at p=21569. [R. J. Mathar, Oct 05 2008]

Differs from A045316 (x^8 == 2 (mod p) has no solution) first at a(37) = 257 which is not a term of A045316. See A070184 for all such terms. - M. F. Hasler, Jun 21 2024

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

MATHEMATICA

ok[p_] := Reduce[Mod[x^32 - 2, p] == 0, x, Integers] == False; Select[Prime[Range[100]], ok ] (* Vincenzo Librandi, Sep 20 2012 *)

PROG

(Magma) [p: p in PrimesUpTo(450) | not exists{x : x in ResidueClassRing(p) | x^32 eq 2 }]; // Vincenzo Librandi, Sep 20 2012

CROSSREFS

Cf. A000040, A049564, A216747.

Cf. A070184 = (this sequence) \ A045316.

KEYWORD

nonn,easy

AUTHOR

Klaus Brockhaus, Jan 27 2001

STATUS

approved

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