A096636 - OEIS

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A096636

Smallest prime p > prime(n+2) such that p is a quadratic residue mod the first n odd primes 3, 5, 7, 11, ..., prime(n+1), and p is a quadratic non-residue mod prime(n+2).

5, 7, 19, 79, 331, 751, 1171, 7459, 10651, 18379, 90931, 78439, 399499, 644869, 2631511, 1427911, 4355311, 5715319, 49196359, 43030381, 163384621, 249623581, 452980999, 1272463669, 505313251 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,1`
	COMMENTS	`Same as smallest prime p with property that the Legendre symbol (p\|q) = 1 for the first n odd primes q = prime(k+1), k = 1, 2, ..., n, and (p\|q) = -1 for q = prime(n+2). - T. D. Noe, Mar 06 2013`
	LINKS	`Table of n, a(n) for n=0..24.`
	EXAMPLE	`Let f(p) = list of Legendre(p\|q) for q = 3,5,7,11,13,...` `Then f(3), f(5), f(7), f(11), ... are:` `p=3: 0, -1, -1, 1, 1, -1, -1, 1, -1, -1, 1, -1, -1, 1, -1, 1, 1, -1, 1, 1, -1, 1, -1, ...` `p=5: -1, 0, -1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, -1, -1, 1, 1, -1, 1, -1, 1, -1, 1, ...` `p=7: 1, -1, 0, -1, -1, -1, 1, -1, 1, 1, 1, -1, -1, 1, 1, 1, -1, -1, -1, -1, -1, 1, -1, ...` `p=11: -1, 1, 1, 0, -1, -1, 1, -1, -1, -1, 1, -1, 1, -1, 1, -1, -1, -1, -1, -1, 1, 1, 1, ...` `p=13: 1, -1, -1, -1, 0, 1, -1, 1, 1, -1, -1, -1, 1, -1, 1, -1, 1, -1, -1, -1, 1, -1, -1, ...` `p=17: -1, -1, -1, -1, 1, 0, 1, -1, -1, -1, -1, -1, 1, 1, 1, 1, -1, 1, -1, -1, -1, 1, 1, ...` `p=19: 1, 1, -1, -1, -1, 1, 0, -1, -1, 1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, -1, -1, ...` `p=5 is the first list that begins with -1, so a(0) = 5,` `p=7 is the first list that begins 1, -1, so a(1) = 7,` `p=19 is the first list that begins 1, 1, -1, so a(2) = 19.`
	MATHEMATICA	`f[n_] := Block[{k = 2}, While[ JacobiSymbol[n, Prime[k]] == 1, k++ ]; Prime[k]]; t = Table[0, {50}]; Do[p = Prime[n]; a = f[p]; If[ t[[ PrimePi[a]]] == 0, t[[ PrimePi[a]]] = p; Print[ PrimePi[a], " = ", p]], {n, 10^9}]`
	CROSSREFS	`Cf. A094929, A222756 (p and q switched).` `See also A096637, A096638, A096639, A096640. - Jonathan Sondow, Mar 07 2013` `Sequence in context: A045447 A159048 A171131 * A101588 A062654 A130729` `Adjacent sequences: A096633 A096634 A096635 * A096637 A096638 A096639`
	KEYWORD	`nonn`
	AUTHOR	`Robert G. Wilson v, Jun 24 2004`
	EXTENSIONS	`Better definition from T. D. Noe, Mar 06 2013` `Entry revised by N. J. A. Sloane, Mar 06 2013` `Simpler definition from Jonathan Sondow, Mar 06 2013`
	STATUS	`approved`

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Last modified June 19 07:51 EDT 2013. Contains 226399 sequences.