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Number of prime numbers of the form k^2 + k + 41 below 10^n.
3

%I #23 Jan 31 2020 20:17:11

%S 0,8,31,86,221,581,1503,4149,11355,31985,90940,261081,756081,2208197,

%T 6483148,19132652,56714624,168806741,504209234

%N Number of prime numbers of the form k^2 + k + 41 below 10^n.

%D Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.

%H Justin DeBenedetto and Jeremy Rouse, <a href="https://arxiv.org/abs/1207.7291">A 60,000 digit prime number of the form x^2 + x + 41</a>, arXiv preprint arXiv:1207.7291 [math.NT] (2012).

%H Henri Cohen, <a href="http://www.math.u-bordeaux.fr/~cohen/hardylw.dvi">High-precision computation of Hardy-Littlewood constants</a>, (1998).

%H Henri Cohen, <a href="/A221712/a221712.pdf">High-precision calculation of Hardy-Littlewood constants</a>. [local copy with permission]

%H Gilbert W. Fung and Hugh C. Williams, <a href="https://doi.org/10.1090/S0025-5718-1990-1023759-3">Quadratic polynomials which have a high density of prime values</a>, Mathematics of Computation, Vol. 55, No. 191 (1990), pp. 345-353.

%H G. H. Hardy and J. E. Littlewood, <a href="https://doi.org/10.1142/9789814542487_0002">Some problems in "Partitio numerorum", III: On the expression of a number as a sum of primes</a>, Acta Mathematica, Vol. 44 (1923), pp. 1-70.

%H R. A. Mollin, <a href="http://www.jstor.org/stable/2975080">Prime-Producing Quadratics</a>, The American Mathematical Monthly, Vol. 104, No. 6 (1997), pp. 529-544.

%H Daniel Shanks, <a href="https://doi.org/10.1090/S0025-5718-1975-0409357-2">Calculation and applications of Epstein zeta functions</a>, Mathematics of Computation, Vol. 29, No. 129 (1975), pp. 271—287.

%H M. L. Stein, S. M. Ulam and M. B. Wells, <a href="http://www.jstor.org/stable/2312588">A Visual Display of Some Properties of the Distribution of Primes</a>, The American Mathematical Monthly, Vol. 71, No. 5 (1964), pp. 516-520.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Ulam_spiral#Hardy_and_Littlewood&#39;s_Conjecture_F">Ulam spiral</a>.

%F 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).

%e 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.

%t 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

%Y Cf. A002837, A005846, A056561, A202018, A221712, A331876.

%K nonn,more

%O 1,2

%A _Amiram Eldar_, Oct 01 2018