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A060380
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Let f(m) = smallest prime that divides k^2 + k + m for k = 0,1,2,...; sequence gives smallest m >= 2 such that f(m) is the n-th prime, or -1 if no such m exists.
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4
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2, 3, 5, 47, 11, 221, 17, 1217, 941, 2747, 8081, 9281, 41, 55661, 19421, 333491, 1262201, 601037, 5237651, 9063641, 12899891, 26149427, 24073871, 28537121, 352031501, 398878547, 160834691, 67374467, 146452961, 24169417397
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OFFSET
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1,1
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COMMENTS
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Chris Nash (see the Prime Puzzles link) has shown that such an m always exists.
For n>2, least odd number d such that the Legendre symbol (1-4d/prime(k)) = -1 for k = 2,...,n, but not for n+1. See A060392. - T. D. Noe, Apr 19 2004
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REFERENCES
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R. F. Lukes, C. D. Patterson and H. C. Williams, Numerical sieving devices: their history and some applications. Nieuw Arch. Wisk. (4) 13 (1995), no. 1, 113-139. Math. Rev. 96m:11082
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LINKS
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EXAMPLE
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k^2 + k + 2 takes the values 2, 4, 8, 14, ... for k = 0,1,2,...; the smallest prime divisor of these numbers is 2, so f(2) = 2.
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MATHEMATICA
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(* This program is not convenient beyond a(24) *) a[1] = 2; a[2] = 3; a[n_] := For[d = 1, True, d = d+2, If[And @@ (# == -1 & /@ Table[JacobiSymbol[1 - 4d, Prime[k]], {k, 2, n}]) && JacobiSymbol[1 - 4d, Prime[n+1]] != -1, Return[d]]]; Table[Print[an = a[n]]; an, {n, 1, 24}] (* Jean-François Alcover, Oct 14 2013, after T. D. Noe *)
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CROSSREFS
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KEYWORD
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hard,nice,nonn
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AUTHOR
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Luis Rodriguez-Torres (ludovicusmagister(AT)yahoo.com), Apr 03 2001
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EXTENSIONS
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STATUS
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approved
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