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

1,1

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).

Links

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

Example

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.

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);

Crossrefs

Cf. A034794, A034795.

Sequence in context: A116968 A162172 A274682 * A339868 A181758 A285790

Adjacent sequences: A359630 A359631 A359632 * A359634 A359635 A359636

Keyword

nonn

Author

Robert Israel, Jan 07 2023

Status

approved