A331876 -id:A331876 - 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).

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).

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

Original entry on oeis.org

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

A319906 Number of prime numbers of the form k^2 + k + 41 below 10^n.

Original entry on oeis.org

0, 8, 31, 86, 221, 581, 1503, 4149, 11355, 31985, 90940, 261081, 756081, 2208197, 6483148, 19132652, 56714624, 168806741, 504209234

Offset: 1

Amiram Eldar, Oct 01 2018

			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.

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.

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.

Cf. A002837, A005846, A056561, A202018, A221712, A331876.

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

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).

A228123 Number of primes generated from Euler's polynomial x^2 + x + 41 from x = 1 to 10^n.

Original entry on oeis.org

1, 10, 86, 581, 4148, 31984, 261080, 2208196, 19132652, 168806740, 1510676802

Offset: 0

Shyam Sunder Gupta, Aug 11 2013

			a(4) = 4148 because the number of primes generated from Euler's polynomial x^2 + x + 41 from x = 1 to 10^4 are 4148.

Cf. A005846, A056561, A331876.

a = 0; n = 1; t = {}; Do[If[PrimeQ[x^2 + x + 41], a = a + 1]; If[Mod[x, n] == 0, n = n*10; AppendTo[t, a]], {x, 1, 1000000000}]; t

a(n) = A331876(n) - 1. - Amiram Eldar, Sep 23 2023

a(10) from Amiram Eldar, Sep 23 2023

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).

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

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A319906 Number of prime numbers of the form k^2 + k + 41 below 10^n.

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A228123 Number of primes generated from Euler's polynomial x^2 + x + 41 from x = 1 to 10^n.

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Mathematica

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