A359633 a(n) is the least prime > a(n-1) such that a(n-1) and a(n) are quadratic residues mod each other.

2, 7, 29, 53, 59, 137, 139, 173, 179, 193, 197, 223, 241, 251, 317, 353, 383, 389, 409, 419, 457, 461, 467, 541, 557, 563, 593, 601, 607, 701, 743, 761, 769, 773, 787, 797, 811, 853, 857, 859, 881, 883, 929, 937, 941, 947, 977, 991, 1009, 1013, 1019, 1033, 1039, 1049, 1051, 1097, 1129, 1153, 1171

Offset: 1

Author

Robert Israel, Jan 07 2023

Keywords

• nonn

Comments

Quadratic reciprocity says that for odd primes p and q, if p is a quadratic residue mod q then q is a quadratic residue mod p except in the case where p and q are both congruent to 3 (mod 4), in which case they can't both be quadratic residues mod each other. Thus if a(n-1) == 1 (mod 4), a(n) is the least prime > a(n-1) that is a quadratic residue mod a(n-1), while if a(n-1) == 3 (mod 4), a(n) is the least prime > a(n-1) that is congruent to 1 (mod 4) and is a quadratic residue mod a(n-1).

Examples

			a(3) = 29 because a(2) = 7, 29 is a quadratic residue mod 7 and 7 is a quadratic residue mod 29, and 29 is the least prime > 7 that works.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A034794, A034795.

Programs

Maple

f:= proc(p) local q;
   q:= p;
   do
     q:= nextprime(q);
     if NumberTheory:-QuadraticResidue(q,p) = 1 and NumberTheory:-QuadraticResidue(p,q) = 1  then return q fi
   od
end proc:
A[1]:= 2: for i from 2 to 100 do A[i]:= f(A[i-1]) od:
seq(A[i], i=1..100);