cp's OEIS Frontend

A088193 Prime numbers where the sequence of largest quadratic residues modulo the primes (A088190) is non-monotonic.

%I A088193 #10 Nov 01 2024 09:36:04
%S A088193 3,7,31,71,103,151,199,239,271,311,359,463,599,719,823,839,911,1063,
%T A088193 1231,1279,1303,1439,1559,1871,1879,1951,1999,2143,2239,2311,2351,
%U A088193 2383,2399,2551,2711,2791,3191,3391,3463,3559,3583,3823,3911,3919,4079,4159
%N A088193 Prime numbers where the sequence of largest quadratic residues modulo the primes (A088190) is non-monotonic.
%C A088193 From the second term on, these primes are always ==7 mod 8. (Tested for the first 20000 primes)
%C A088193 From _Robert Israel_, Oct 31 2024: (Start)
%C A088193 This is true because if prime(n) == 1 mod 4, A088190(n) = prime(n) - 1 while if prime(n) == 3 mod 8, A088190(n) = prime(n) - 2.  In either case, A088190(n) > prime(n-1) - 1 >= A088190(n-1).
%C A088193 Primes prime(n) such that A088190(n) <= A088190(n-1). (End)
%H A088193 Robert Israel, <a href="/A088193/b088193.txt">Table of n, a(n) for n = 1..10000</a>
%p A088193 lqr:= proc(p) local k;
%p A088193   for k from p-1 by -1 do if numtheory:-quadres(k,p) = 1 then return k fi od:
%p A088193 end proc:
%p A088193 p:= 2: v:= lqr(2): R:= NULL: count:= 0:
%p A088193 while count < 100 do
%p A088193   q:= p; vq:= v; p:= nextprime(p); v:= lqr(p);
%p A088193   if v <= vq then R:= R,p; count:= count+1;
%p A088193   fi
%p A088193 od:
%p A088193 R; # _Robert Israel_, Oct 31 2024
%o A088193 (PARI) qrp_p_nm(to)= {/* The primes where the sequence of the largest QR modulo the primes is non-monotonic */ local(m,k=1,p,v=[]); for(i=2,to,m=1; p=prime(i); j=2; while((j<=(p-1)/2)&&(m<p-1),m=max(m,(j^2)%p); j++); if((m-k)<=0,v=concat(v,p)); k=m); print(v) }
%Y A088193 Cf. A088190, A088191, A088192, A088194, A088195.
%K A088193 easy,nonn
%O A088193 1,1
%A A088193 Ferenc Adorjan (fadorjan(AT)freemail.hu), Sep 22 2003