A014662 Primes p such that order of 2 mod p (=A007733(p)) is even.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Apart from the first term, identical to A091317. - Charles R Greathouse IV, Feb 13 2009

Dirichlet density is 5/24 (Fein, Gordon, & Smith); they show a result on expressing -1 as the sum of two squares relating to this sequence. - Charles R Greathouse IV, May 15 2024

References

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

Links

• Klaus Brockhaus, Table of n, a(n) for n = 1..1000
• Burton Fein, Basil Gordon, and John H. Smith, On the representation of -1 as a sum of two squares in an algebraic number field J. Num. Theor. (1971) Vol. 3, Issue 3, 310-315.
• Eugen J. Ionascu, Florian Luca, and Thomas Merino, On the average value of the minimal Hamming multiple, arXiv:2412.10839 [math.NT], 2024. See pp. 4, 17.
• C. Smyth, The terms in Lucas Sequences divisible by their indices, JIS 13 (2010) #10.2.4.

Crossrefs

The prime terms of A296243.

Cf. A091317.

Programs

Magma

[ p: p in PrimesInInterval(3, 419) | IsEven(Modorder(2, p)) ]; // Klaus Brockhaus, Dec 09 2008

Maple

select(t -> isprime(t) and numtheory:-order(2,t)::even, [2*i+1 $ i=1..1000]); # Robert Israel, Aug 12 2014

Mathematica

Select[Prime[Range[80]], EvenQ[MultiplicativeOrder[2, #/(2^IntegerExponent[ #, 2])]]&] (* Jean-François Alcover, Sep 02 2018 *)

PARI

isok(p) = isprime(p) && !(znorder(Mod(2, p/2^valuation(p, 2))) % 2); \\ Michel Marcus, Sep 02 2018

PARI

is(n)=n>2 && Mod(2,n)^(n>>valuation(n-1,2))!=1 && isprime(n) \\ Charles R Greathouse IV, May 07 2024

Extensions

More terms from Klaus Brockhaus, Dec 09 2008