A115591 Primes p such that the multiplicative order of 2 modulo p is (p-1)/2.

7, 17, 23, 41, 47, 71, 79, 97, 103, 137, 167, 191, 193, 199, 239, 263, 271, 311, 313, 359, 367, 383, 401, 409, 449, 463, 479, 487, 503, 521, 569, 599, 607, 647, 719, 743, 751, 761, 769, 809, 823, 839, 857, 863, 887, 929, 967, 977, 983, 991, 1009, 1031

Offset: 1

Author

Don Reble, Mar 11 2006

Keywords

• nonn

Comments

It appears that this is also the sequence of values of n for which the sum of terms of one period of the base-2 MR-expansion (see A136042) of 1/n equals (n-1)/2. An example appears in A155072 where one period of the base-2 MR-expansion of 1/17 is shown to be {5,1,1,1} with sum 8=(17-1)/2. - John W. Layman, Jan 19 2009

If p is a term of this sequence, then 2 is a quadratic residue module p, so p == 1, 7 (mod 8). - Jianing Song, Nov 01 2024

Links

• Klaus Brockhaus, Table of n, a(n) for n=1..1000

Crossrefs

Cf. A001122, A001133.

Cf. A136042, A155072. - John W. Layman, Jan 19 2009

Programs

Magma

[ p: p in PrimesUpTo(1031) | r eq 1 and Order(R!2) eq q where q,r is Quotrem(p,2) where R is ResidueClassRing(p) ]; // Klaus Brockhaus, Dec 02 2008

Mathematica

fQ[n_] := 1 + 2 MultiplicativeOrder[2, n] == n; Select[ Prime@ Range@ 174, fQ]

PARI

r=2;forprime(p=3,1500,z=(p-1)/znorder(Mod(r,p));if(z==2,print1(p,", "))); \\ Joerg Arndt, Jan 12 2011