A331877 a(n) = number of primes of the form P(k) = k^2 + k + 41 for k <= 10^n as predicted by the Hardy and Littlewood Conjecture F, rounded to nearest integer. The actual number of primes is A331876(n).

17, 97, 586, 4133, 31965, 261022, 2207375, 19129225, 168807923, 1510681420, 13671046376, 124849864598, 1148859448601, 10639680705031, 99077207876785

Offset

1,1

References

See A319906.

Links

Table of n, a(n) for n=1..15.

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.

Michael J. Jacobson, Jr. and Hugh C. Williams, New Quadratic Polynomials With High Densities Of Prime Values, Math. Comp., 72, 241, 499-519, 2002.

Formula

b(m) = round (C * Integral_{x=2..m} x/log(x) dx), with C ~= 3.319773177471..., the Hardy-Littlewood constant for k^2+k+41 (A221712); a(n) = b(10^n).

Prog

(PARI) C=3.31977317747142166532355685764988796646855; for(n=1, 15, print1(round(C*intnum(x=2, 10^n, 1/log(x))), ", "))

Crossrefs

Cf. A221712, A319906, A331876.

Sequence in context: A008514 A152913 A184327 * A358572 A262207 A282997

Adjacent sequences: A331874 A331875 A331876 * A331878 A331879 A331880

Keyword

nonn,more

Author

Hugo Pfoertner, Jan 30 2020

Status

approved