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Search: id:A088190
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| A088190 |
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Largest quadratic residue modulo prime(n). |
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11
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| 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; listen)
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OFFSET
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1,3
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COMMENT
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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.
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LINKS
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F. Adorjan, The sequence of largest quadratic residues modulo the primes.
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FORMULA
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a(n) = max(r, r==j^2 mod p(n)|j=1, 2, ...(p(n)-1)/2)
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PROGRAM
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(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) }
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CROSSREFS
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Cf. A088191, A088192, A088193, A088194, A088195, A088196, A088197, A088198, A088199, A088200, A088201.
Sequence in context: A008794 A075709 A116682 this_sequence A092322 A050218 A165996
Adjacent sequences: A088187 A088188 A088189 this_sequence A088191 A088192 A088193
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KEYWORD
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nonn
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AUTHOR
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Ferenc Adorjan (fadorjan(AT)freemail.hu), Sep 22 2003
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