cp's OEIS Frontend

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

%I A163183 #31 Jun 30 2025 04:36:36
%S A163183 3,11,19,43,59,67,83,107,131,139,163,179,211,227,251,281,283,307,331,
%T A163183 347,379,419,443,467,491,499,523,547,563,571,587,617,619,643,659,683,
%U A163183 691,739,787,811,827,859,883,907,947,971,1019,1033,1049,1051,1091,1097
%N A163183 Primes dividing 2^j + 1 for some odd j.
%C A163183 Also the primes p for which ord_p(-2) is odd, as (-2)^j == 1 (mod p).
%C A163183 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).
%C A163183 Claim: Sequence contains all primes = 3 mod 8, so contains A007520 as a subsequence.
%C A163183 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.
%C A163183 Also contains some primes == 1 (mod 8), given in A163184. So sequence is a union of A007520 and A163184.
%C A163183 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), ....
%C A163183 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.
%C A163183 From _Jianing Song_, Jun 22 2025: (Start)
%C A163183 The multiplicative order of -2 modulo a(n) is A385228(n).
%C A163183 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).
%C A163183 Conjecture: this sequence has density 7/24 among the primes (see A014663). (End)
%H A163183 Jianing Song, <a href="/A163183/b163183.txt">Table of n, a(n) for n = 1..10000</a>
%e A163183 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.
%p A163183 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;
%t A163183 Select[Prime[Range[200]], OddQ[MultiplicativeOrder[-2, #]] &] (* _Paolo Xausa_, Jun 30 2025 *)
%o A163183 (PARI) lista(nn) = forprime(p=3, nn, if(znorder(Mod(-2, p))%2, print1(p, ", "))); \\ _Jinyuan Wang_, Mar 23 2020
%Y A163183 Sequence is a union of A007520 and A163184.
%Y A163183 Subsequence of A033200. Contains A007520 as a subsequence.
%Y A163183 Cf. A385228 (the actual multiplicative orders).
%Y A163183 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).
%K A163183 nonn,easy
%O A163183 1,1
%A A163183 _Christopher J. Smyth_, Jul 22 2009