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

Author

Hugo Pfoertner, Jan 30 2020

Keywords

• nonn
• more

References

• See A319906.

Links

• 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.

Crossrefs

Cf. A221712, A319906, A331876.

Programs

PARI

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

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).