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A319906
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Number of prime numbers of the form k^2 + k + 41 below 10^n.
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3
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0, 8, 31, 86, 221, 581, 1503, 4149, 11355, 31985, 90940, 261081, 756081, 2208197, 6483148, 19132652, 56714624, 168806741, 504209234
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OFFSET
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1,2
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REFERENCES
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Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
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LINKS
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FORMULA
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According to Hardy and Littlewood's Conjecture F: a(n) ~ 2 * C * 10^(n/2)/(n*log(10)), where C = 3.319773... (Hardy-Littlewood constant for x^2+x+41, A221712).
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EXAMPLE
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The first 8 values of k^2 + k + 41 for k = 0 to 7 are above 10 and below 100: 41, 43, 47, 53, 61, 71, 83, 97, thus a(1) = 0 and a(2) = 8.
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MATHEMATICA
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f[n_] := n^2 + n + 41; c = 0; k = 0; a={}; Do[f1 = f[k]; While[f1 < 10^n, If[PrimeQ[f1], c++]; k++; f1 = f[k]]; AppendTo[a, c], {n, 1, 10}]; a
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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