cp's OEIS Frontend

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).

%I A331877 #15 Jan 17 2024 04:31:27
%S A331877 17,97,586,4133,31965,261022,2207375,19129225,168807923,1510681420,
%T A331877 13671046376,124849864598,1148859448601,10639680705031,99077207876785
%N 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).
%D A331877 See A319906.
%H A331877 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 A331877 Michael J. Jacobson, Jr. and Hugh C. Williams, <a href="http://www.ams.org/mcom/2003-72-241/S0025-5718-02-01418-7/S0025-5718-02-01418-7.pdf">New Quadratic Polynomials With High Densities Of Prime Values</a>, Math. Comp., 72, 241, 499-519, 2002.
%F A331877 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).
%o A331877 (PARI) C=3.31977317747142166532355685764988796646855; for(n=1,15,print1(round(C*intnum(x=2,10^n,1/log(x))),", "))
%Y A331877 Cf. A221712, A319906, A331876.
%K A331877 nonn,more
%O A331877 1,1
%A A331877 _Hugo Pfoertner_, Jan 30 2020