%I #42 Nov 23 2024 11:12:14

%S 7,23,31,47,71,73,79,89,103,127,151,167,191,199,223,233,239,263,271,

%T 311,337,359,367,383,431,439,463,479,487,503,599,601,607,631,647,719,

%U 727,743,751,823,839,863,881,887,911,919,937,967,983,991,1031,1039,1063

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

%C Or, primes p which do not divide 2^n+1 for any n.

%C 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

%C 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

%C Has density 7/24 (Hasse).

%D 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.

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

%H T. D. Noe, <a href="/A014663/b014663.txt">Table of n, a(n) for n=1..1000</a>

%H H. H. Hasse, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002296616">Über die Dichte der Primzahlen p, ... </a>, Math. Ann., 168 (1966), 19-23.

%H J. C. Lagarias, <a href="http://projecteuclid.org/euclid.pjm/1102706452">The set of primes dividing the Lucas numbers has density 2/3</a>, Pacific J. Math., 118. No. 2, (1985), 449-461.

%H Chunlei Li, Nian Li, and Matthew G. Parker, <a href="http://www.ii.uib.no/~matthew/Pairs.pdf">Complementary Sequence Pairs of Types II and III</a>. [From _N. J. A. Sloane_, Jun 16 2012]

%t okQ[p_] := OddQ[MultiplicativeOrder[2, p]];

%t Select[Prime[Range[1000]], okQ] (* _Jean-François Alcover_, Nov 23 2024 *)

%o (PARI) isA014663(p)=1==Mod(1,p)<<((p-1)>>factor(p-1,2)[1,2])

%o listA014663(N=1000)=forprime(p=3,N,isA014663(p)&print1(p", ")) \\ _M. F. Hasler_, Dec 08 2007

%o (PARI) lista(nn) = {forprime(p=3, nn, if (znorder(Mod(2, p)) % 2, print1(p, ", ")););} \\ _Michel Marcus_, Feb 06 2015

%Y Cf. Complement in primes of A091317.

%Y Cf. A040098, A045315, A049564.

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

%K nonn,changed

%O 1,1

%A _N. J. A. Sloane_, Dec 11 1999.

%E Edited by _M. F. Hasler_, Dec 08 2007

%E More terms from _Max Alekseyev_, Feb 06 2010