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A359633
a(n) is the least prime > a(n-1) such that a(n-1) and a(n) are quadratic residues mod each other.
1
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
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
Sequence in context: A116968 A162172 A274682 * A339868 A181758 A285790
KEYWORD
nonn
AUTHOR
Robert Israel, Jan 07 2023
STATUS
approved