The OEIS is supported by the many generous donors to the OEIS Foundation.

A088192
Distance between prime(n) and the largest quadratic residue modulo prime(n).
14
1, 2, 1, 3, 2, 1, 1, 2, 5, 1, 3, 1, 1, 2, 5, 1, 2, 1, 2, 7, 1, 3, 2, 1, 1, 1, 3, 2, 1, 1, 3, 2, 1, 2, 1, 3, 1, 2, 5, 1, 2, 1, 7, 1, 1, 3, 2, 3, 2, 1, 1, 7, 1, 2, 1, 5, 1, 3, 1, 1, 2, 1, 2, 11, 1, 1, 2, 1, 2, 1, 1, 7, 3, 1, 2, 5, 1, 1, 1, 1, 2, 1, 7, 1, 3, 2, 1, 1, 1, 3, 2, 13, 3, 2, 2, 5, 1, 1, 2, 1
OFFSET
1,2
a(n) = smallest m>0 such that -m is a quadratic residue modulo prime(n).
a(n) = smallest m>0 such that prime(n) either splits or ramifies in the imaginary quadratic field Q(sqrt(-m)). Equals -A220862(n) except when n = 1. Cf. A220861, A220863. - N. J. A. Sloane, Dec 26 2012
The values are 1 or a prime number (easily provable!). The maximum occurring prime values increase very slowly: up to 10^5 terms the largest prime is 43. The primes do not appear in order.
REFERENCES
David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, Cor. 5.17, p. 105. - From N. J. A. Sloane, Dec 26 2012
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
J. A. Bergstra, I. Bethke, A negative result on algebraic specifications of the meadow of rational numbers, arXiv preprint arXiv:1507.00548 [math.RA], 2015-2016.
FORMULA
a(n) = A053760(n) unless -1 is a quadratic residue mod prime(n). - Charles R Greathouse IV, Oct 31 2012
MATHEMATICA
a[n_] := With[{p = Prime[n]}, If[JacobiSymbol[-1, p] > 0, 1, For[d = 2, True, d = NextPrime[d], If[JacobiSymbol[-d, p] >= 0, Return[d]]]]]; Array[a, 100] (* Jean-François Alcover, Feb 16 2018, after Charles R Greathouse IV *)
PROG
(PARI) qrp_pm(fr, to)= {/* The distance of largest QR modulo the primes from the primes */ local(m, p, v=[]); for(i=fr, to, m=1; p=prime(i); j=2; while((j<=(p-1)/2)&&(m<p-1), m=max(m, (j^2)%p); j++); v=concat(v, p-m)); print(v) }
(PARI) do(p)=if(kronecker(-1, p)>0, 1, forprime(d=2, p, if(kronecker(-d, p) >= 0, return(d))))
apply(do, primes(100)) \\ Charles R Greathouse IV, Oct 31 2012
CROSSREFS
Records are (essentially) given by A147971.
Sequence in context: A324607 A023512 A322027 * A218459 A056062 A230517
KEYWORD
easy,nonn
AUTHOR