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A083844
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Number of primes of the form x^2 + 1 < 10^n.
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18
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2, 4, 10, 19, 51, 112, 316, 841, 2378, 6656, 18822, 54110, 156081, 456362, 1339875, 3954181, 11726896, 34900213, 104248948, 312357934, 938457801, 2826683630, 8533327397, 25814570672, 78239402726, 237542444180, 722354138859, 2199894223892
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OFFSET
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1,1
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COMMENTS
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It is conjectured that there are infinitely many primes of the form x^2 + 1 (and thus this sequence never becomes constant), but this has not been proved.
These primes can be found quickly using a sieve based on the fact that numbers of this form have at most one primitive prime factor (A005529). The sum of the reciprocals of these primes is 0.81459657... - T. D. Noe, Oct 14 2003
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REFERENCES
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G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 17.
P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 190.
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LINKS
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EXAMPLE
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a(3) = 10 because the only primes or the form x^2 + 1 < 10^3 are the ten primes: 2, 5, 17, 37, 101, 197, 257, 401, 577, 677.
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MATHEMATICA
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c = 1; k = 2; (* except for the initial prime 2, all X's must be odd. *) Do[ While[ k^2 + 1 < 10^n, If[ PrimeQ[k^2 + 1], c++ ]; k += 2]; Print[c], {n, 1, 20}]
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CROSSREFS
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Cf. A005529 (primitive prime factors of the sequence k^2+1).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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a(23)-a(25) from Marek Wolf and Robert Gerbicz (code from Robert, computation done by Marek) Robert Gerbicz, Mar 13 2010
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STATUS
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approved
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