A319906 -id:A319906 - cp's OEIS frontend

A221712 Hardy-Littlewood constant for x^2+x+41.

3, 3, 1, 9, 7, 7, 3, 1, 7, 7, 4, 7, 1, 4, 2, 1, 6, 6, 5, 3, 2, 3, 5, 5, 6, 8, 5, 7, 6, 4, 9, 8, 8, 7, 9, 6, 6, 4, 6, 8, 5, 5, 4, 5, 8, 5, 6, 5, 2, 9, 8, 5, 8, 4, 9, 1, 5, 3, 9, 4, 0, 7, 2, 7, 9, 5, 0, 2, 6, 3, 3, 1, 0, 4, 2, 6, 1, 1, 8, 1, 4, 9, 7, 3, 7, 5, 5

Offset: 1

N. J. A. Sloane, Jan 26 2013

			3.31977317747142166532355685764988796646855...

Henri Cohen, Number Theory, Vol II: Analytic and Modern Tools, Springer (Graduate Texts in Mathematics 240), 2007.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 265-266.

Christian Axler and Mehdi Hassani, Carleman's inequality over prime numbers, Integers (2021) Vol. 21, #A53.
Karim Belabas and Henri Cohen, Computation of the Hardy-Littlewood constant for quadratic polynomials, PARI/GP script, 2020.
Lea Beneish and Christopher Keyes, How often does a cubic hypersurface have a rational point?, arXiv:2405.06584 [math.NT], 2024. See p. 23.
David Broadhurst, Five families of rapidly convergent evaluations of zeta values, arXiv:2401.08997 [math.NT], 2024.
Henri Cohen, High-precision calculation of Hardy-Littlewood constants, (1998).
Henri Cohen, High precision computation of Hardy-Littlewood constants. [Cached pdf version, with permission]
Stéphane Vinatier and William Reginald Alcorn, Mathematics as seen by an artist: Inspiring mathematical objects, Eur. Math. Soc. Mag. (2025) Vol. 135, 20-31. See p. 26.

Cf. A005846, A056561, A202018, A221713, A319906, A331876, A331877.

\\ See Belabas, Cohen link. Run as HardyLittlewood2(x^2+x+41)/2 after setting the required precision.

More terms from Hugo Pfoertner, Jan 31 2020

A331876 Number of primes of the form P(k) = k^2 + k + 41 for k <= 10^n, where P(k) is Euler's prime-generating polynomial A202018.

Original entry on oeis.org

2, 11, 87, 582, 4149, 31985, 261081, 2208197, 19132653, 168806741, 1510676803

Offset: 0

Hugo Pfoertner, Jan 30 2020

			a(0) = 2 because 41 and 43 are the 2 primes generated for k <= 1 = 10^0.
a(1) = 11 because 41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151 are the 11 primes generated for k <= 10^1, (A202018(10) = 151).
a(3) = 87 because 87 terms of A202018(0..100) are prime. The 14 composites occur for k = A007634(1..14): 40, 41, 44, 49, 56, ...

Cf. A005846, A007634, A202018, A319906, A331877.

n=0;m=1;for(k=0,10^7,my(j=k^2+k+41);if(isprime(j),n++);if(k==m,m*=10;print1(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).

Original entry on oeis.org

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

Offset: 1

Hugo Pfoertner, Jan 30 2020

See A319906.

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.

Cf. A221712, A319906, A331876.

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

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).

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A221712 Hardy-Littlewood constant for x^2+x+41.

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A331876 Number of primes of the form P(k) = k^2 + k + 41 for k <= 10^n, where P(k) is Euler's prime-generating polynomial A202018.

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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).

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