OFFSET
1,3
COMMENTS
Denote a(n) by LQR(p_n). Observations (tested up to 20000 primes): - the sequence of largest QR modulo the primes (LQR(p_n) is 'almost' monotonic, - p_n-LQR(p_n) is either 1 or a prime value (see A088192) - if LQR(p_n)<=LQR(p_{n-1}) then p_n==7 mod 8 (when n>2) (see A088194) - if LQR(p_n)<=LQR(p_{n-1}) then p_n-LQR(p_n) is an odd prime, but never 5 (see A088195) For a similar set of sequences, related to quadratic non-residues, see A088196-A088201.
From Robert Israel, Oct 31 2024: (Start)
a(n) = prime(n)-1 if and only if n is 1 or in A080147.
a(n) = prime(n)-2 if and only if prime(n) is in A007520.
a(n) = prime(n)-3 if and only if prime(n) is in A107006. (End)
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = max(r, r==j^2 mod p(n)|j=1, 2, ...(p(n)-1)/2)
MAPLE
lqr:= proc(p) local k;
for k from p-1 by -1 do if numtheory:-quadres(k, p) = 1 then return k fi od:
end proc:
seq(lqr(ithprime(i)), i=1..100); # Robert Israel, Oct 31 2024
MATHEMATICA
a[n_] := With[{p = Prime[n]}, SelectFirst[Range[p - 1, 1, -1], JacobiSymbol[#, p] == 1&]]; Array[a, 100] (* Jean-François Alcover, Feb 16 2018 *)
PROG
(PARI) qrp(fr, to)= {/* Sequence of the largest QR modulo 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, m)); print(v) }
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Ferenc Adorjan (fadorjan(AT)freemail.hu), Sep 22 2003
STATUS
approved