A001134 Primes p such that the multiplicative order of 2 modulo p is (p-1)/4.

113, 281, 353, 577, 593, 617, 1033, 1049, 1097, 1153, 1193, 1201, 1481, 1601, 1889, 2129, 2273, 2393, 2473, 3049, 3089, 3137, 3217, 3313, 3529, 3673, 3833, 4001, 4217, 4289, 4457, 4801, 4817, 4937, 5233, 5393, 5881, 6121, 6521, 6569, 6761, 6793, 6841, 7129, 7481, 7577, 7793, 7817, 7841, 8209

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

The multiplicative order of x modulo y is the smallest positive number m such that x^m is congruent to 1 mod y.

References

• M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 59.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

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

Crossrefs

Cf. A001133, A001135, A001136, A115586, A115591.

Programs

Magma

[ p: p in PrimesUpTo(6761) | r eq 1 and Order(R!2) eq q where q,r is Quotrem(p,4) where R is ResidueClassRing(p) ]; // Klaus Brockhaus, Dec 02 2008

Mathematica

Reap[For[p = 2, p <= 6761, p = NextPrime[p], If[ MultiplicativeOrder[2, p] == (p-1)/4, Sow[p]]]][[2, 1]] (* Jean-François Alcover, May 17 2013 *)

PARI

forprime(p=3,10^4,if(znorder(Mod(2,p))==(p-1)/4,print1(p,", "))); \\ Joerg Arndt, May 17 2013

PARI

oddres(n)=n>>valuation(n, 2)
cyc(d)=my(k=1, t=1,y=(d-5)/(2*3)+1); while((t=oddres(t+d))>1 && k<=y, k++); k
forstep(n=1, 241537, [16,8], if(cyc(n)==n>>3,print1(n", "))) ; \\ Charles R Greathouse IV, May 18 2013

Extensions

More terms and better definition from Don Reble, Mar 11 2006