A001134 Primes p such that the multiplicative order of 2 modulo p is (p-1)/4. (Formerly M5371 N2332)

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

1,1

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

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A325067 A142641 A070181 * A070188 A059331 A124586

Adjacent sequences: A001131 A001132 A001133 * A001135 A001136 A001137

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved