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A088190 Largest quadratic residue modulo prime(n). 12
1, 1, 4, 4, 9, 12, 16, 17, 18, 28, 28, 36, 40, 41, 42, 52, 57, 60, 65, 64, 72, 76, 81, 88, 96, 100, 100, 105, 108, 112, 124, 129, 136, 137, 148, 148, 156, 161, 162, 172, 177, 180, 184, 192, 196, 196, 209, 220, 225, 228, 232, 232, 240, 249, 256, 258, 268, 268, 276 (list; graph; refs; listen; history; text; internal format)
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.

LINKS

Table of n, a(n) for n=1..59.

F. Adorjan, The sequence of largest quadratic residues modulo the primes.

FORMULA

a(n) = max(r, r==j^2 mod p(n)|j=1, 2, ...(p(n)-1)/2)

MATHEMATICA

a[n_] := With[{p = Prime[n]}, SelectFirst[Range[p - 1, 1, -1], JacobiSymbol[#, p] == 1&]]; Array[a, 100] (* Jean-Fran├žois 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

Cf. A088191, A088192, A088193, A088194, A088195, A088196, A088197, A088198, A088199, A088200, A088201.

Sequence in context: A116682 A168157 A222045 * A092322 A050218 A165996

Adjacent sequences:  A088187 A088188 A088189 * A088191 A088192 A088193

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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Last modified October 1 17:45 EDT 2020. Contains 337444 sequences. (Running on oeis4.)