A163183 Primes dividing 2^j + 1 for some odd j.

3, 11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251, 281, 283, 307, 331, 347, 379, 419, 443, 467, 491, 499, 523, 547, 563, 571, 587, 617, 619, 643, 659, 683, 691, 739, 787, 811, 827, 859, 883, 907, 947, 971, 1019, 1033, 1049, 1051, 1091, 1097

Offset: 1

Author

Christopher J. Smyth, Jul 22 2009

Keywords

• nonn
• easy

Comments

Also the primes p for which ord_p(-2) is odd, as (-2)^j == 1 (mod p).

All such p are = 1 or 3 mod 8, so sequence is subsequence of A033200, as (-2)^{j+1} == -2 (mod p) implies that (-2/p) = 1, p == 1 or 3 (mod 8).

Claim: Sequence contains all primes = 3 mod 8, so contains A007520 as a subsequence.

Proof: If p = 8r + 3 then 2^{4r+1} == 1 or -1 (mod p). If former, then (2^{2r+1})^2 == 2 (mod p), (2/p) = 1, only true for p == 1 or 7 (mod 8). So p | 2^{4r+1} + 1.

Also contains some primes == 1 (mod 8), given in A163184. So sequence is a union of A007520 and A163184.

Claim: For every p in sequence and every 2^k, the equation x^{2^k} == -2 (mod p) is soluble. Hence sequence is a subsequence of A033203 (k=1), A051071 (k=2), A051073 (k=3), A051077 (k=4), A051085 (k=5), A051101 (k=6), ....

Proof: Put x == (-2)^u (mod p). Then using (-2)^j == 1 (mod p), we can solve x^{2^k} == -2 (mod p) if can find u and v such that u*2^k + v*j = 1, possible as gcd(2^k, j) = 1.

From Jianing Song, Jun 22 2025: (Start)

The multiplicative order of -2 modulo a(n) is A385228(n).

Contained in primes congruent to 1 or 3 modulo 8 (primes p such that -2 is a quadratic residue modulo p, A033200), and contains primes congruent to 3 modulo 8 (A007520).

Conjecture: this sequence has density 7/24 among the primes (see A014663). (End)

Examples

			11 is in sequence as 11 | 2^5 + 1; 281 (smallest element of the sequence == 1 (mod 8)) is in the sequence as 281 | 2^35 + 1.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..10000

Crossrefs

Sequence is a union of A007520 and A163184.

Subsequence of A033200. Contains A007520 as a subsequence.

Cf. A385228 (the actual multiplicative orders).

Cf. other bases: A014663 (base 2), A385220 (base 3), A385221 (base 4), A385192 (base 5), this sequence (base -2), A385223 (base -3), A385224 (base -4), A385225 (base -5).

Programs

Maple

with(numtheory):A:=3:p:=3: for c to 500 do p:=nextprime(p);if order(-2,p) mod 2=1 then A:=A,p;;fi;od:A;

Mathematica

Select[Prime[Range[200]], OddQ[MultiplicativeOrder[-2, #]] &] (* Paolo Xausa, Jun 30 2025 *)

PARI

lista(nn) = forprime(p=3, nn, if(znorder(Mod(-2, p))%2, print1(p, ", "))); \\ Jinyuan Wang, Mar 23 2020