A307965 a(n) is the least prime p = prime(k) > prime(n) such that A306530(k) = prime(n).

7, 11, 19, 53, 43, 173, 67, 2477, 8803, 9173, 32323, 37123, 163, 74093, 170957, 360293, 679733, 2404147, 2004917, 69009533, 51599563, 155757067, 96295483, 146161723, 1408126003, 3519879677, 2050312613, 3341091163, 78864114883, 65315700413, 1728061733, 9447241877

Offset: 1

Author

Amiram Eldar and Thomas Ordowski, May 08 2019

Keywords

• nonn

Comments

This sequence is analogous to A000229, but for least prime quadratic residue modulo p.

Note that a(n) is the least odd number m > prime(n) such that prime(n)^((m-1)/2) == 1 (mod m) and q^((m-1)/2) == -1 (mod m) for every prime q < prime(n). Such m is always an odd prime.

Crossrefs

Cf. A000229, A306530.

Programs

Mathematica

f[n_] := Module[{p = Prime[n], q = 2}, While[JacobiSymbol[q, p] != 1, q = NextPrime[q]]; q]; a[n_] := Module[{p = Prime[n], k = n + 1}, While[f[k] != p, k++]; Prime[k]]; Array[a, 20]

PARI

f(n) = my(i=1, p = prime(n)); while(kronecker(prime(i), p)! = 1, i++); prime(i); \\ A306530
a(n) = my(p=prime(n), iq = p+1, q=nextprime(iq)); while(f(iq)!= p, iq++); prime(iq); \\ Michel Marcus, May 12 2019