

A083844


Number of primes of the form x^2 + 1 < 10^n.


18



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

1,1


COMMENTS

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


REFERENCES

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. SpringerVerlag, 1991, p. 190.


LINKS



EXAMPLE

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.


MATHEMATICA

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}]


CROSSREFS

Cf. A005529 (primitive prime factors of the sequence k^2+1).


KEYWORD

nonn


AUTHOR



EXTENSIONS

a(23)a(25) from Marek Wolf and Robert Gerbicz (code from Robert, computation done by Marek) Robert Gerbicz, Mar 13 2010


STATUS

approved



