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

3, 7, 31, 71, 103, 151, 199, 239, 271, 311, 359, 463, 599, 719, 823, 839, 911, 1063, 1231, 1279, 1303, 1439, 1559, 1871, 1879, 1951, 1999, 2143, 2239, 2311, 2351, 2383, 2399, 2551, 2711, 2791, 3191, 3391, 3463, 3559, 3583, 3823, 3911, 3919, 4079, 4159

Offset

1,1

Comments

From the second term on, these primes are always ==7 mod 8. (Tested for the first 20000 primes)

From Robert Israel, Oct 31 2024: (Start)

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).

Primes prime(n) such that A088190(n) <= A088190(n-1). (End)

Links

Robert Israel, Table of n, a(n) for n = 1..10000

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:
p:= 2: v:= lqr(2): R:= NULL: count:= 0:
while count < 100 do
  q:= p; vq:= v; p:= nextprime(p); v:= lqr(p);
  if v <= vq then R:= R, p; count:= count+1;
  fi
od:
R; # Robert Israel, Oct 31 2024

Prog

(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) }

Crossrefs

Cf. A088190, A088191, A088192, A088194, A088195.

Sequence in context: A292890 A042131 A109140 * A091383 A166501 A261862

Adjacent sequences: A088190 A088191 A088192 * A088194 A088195 A088196

Keyword

easy,nonn

Author

Ferenc Adorjan (fadorjan(AT)freemail.hu), Sep 22 2003

Status

approved