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

1,1

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)

Links

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

Example

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.

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 *)

Prog

(PARI) lista(nn) = forprime(p=3, nn, if(znorder(Mod(-2, p))%2, print1(p, ", "))); \\ Jinyuan Wang, Mar 23 2020
(Python)
from sympy import n_order, nextprime
from itertools import islice
def A163183_gen(): # generator of terms
    p = 3
    while True:
        if n_order(-2, p)&1:
            yield p
        p = nextprime(p)
A163183_list = list(islice(A163183_gen(), 20)) # Chai Wah Wu, Feb 05 2026

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).

Sequence in context: A161429 A079544 A192717 * A007520 A294912 A309027

Adjacent sequences: A163180 A163181 A163182 * A163184 A163185 A163186

Keyword

nonn,easy

Author

Christopher J. Smyth, Jul 22 2009

Status

approved