A319906 Number of prime numbers of the form k^2 + k + 41 below 10^n.
0, 8, 31, 86, 221, 581, 1503, 4149, 11355, 31985, 90940, 261081, 756081, 2208197, 6483148, 19132652, 56714624, 168806741, 504209234
Offset: 1
Examples
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.
References
- Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
Links
- Justin DeBenedetto and Jeremy Rouse, A 60,000 digit prime number of the form x^2 + x + 41, arXiv preprint arXiv:1207.7291 [math.NT] (2012).
- Henri Cohen, High-precision computation of Hardy-Littlewood constants, (1998).
- Henri Cohen, High-precision calculation of Hardy-Littlewood constants. [local copy with permission]
- Gilbert W. Fung and Hugh C. Williams, Quadratic polynomials which have a high density of prime values, Mathematics of Computation, Vol. 55, No. 191 (1990), pp. 345-353.
- G. H. Hardy and J. E. Littlewood, Some problems in "Partitio numerorum", III: On the expression of a number as a sum of primes, Acta Mathematica, Vol. 44 (1923), pp. 1-70.
- R. A. Mollin, Prime-Producing Quadratics, The American Mathematical Monthly, Vol. 104, No. 6 (1997), pp. 529-544.
- Daniel Shanks, Calculation and applications of Epstein zeta functions, Mathematics of Computation, Vol. 29, No. 129 (1975), pp. 271—287.
- M. L. Stein, S. M. Ulam and M. B. Wells, A Visual Display of Some Properties of the Distribution of Primes, The American Mathematical Monthly, Vol. 71, No. 5 (1964), pp. 516-520.
- Wikipedia, Ulam spiral.
Programs
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Mathematica
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
Formula
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).