A394750 Integers k such that x^2 + x - k attains a record Hardy-Littlewood constant.

1, 43, 73, 109, 283, 619, 2293, 6163, 7759, 23929, 42739, 56269, 82009, 90073, 153409, 169933, 211999, 249439, 321889, 328063, 349513, 501229

Offset

1,2

Comments

The Hardy-Littlewood conjecture predicts an asymptotic density for the primes generated by the quadratic polynomial P_k(x) = x^2 + x + k. The associated Hardy-Littlewood constant is given by the Euler product C(k) = Product_{primes p} (1 - w_p/p) / (1 - 1/p), where w_p is the number of solutions of P_k(x) = 0 (mod p).

For odd primes p, one has w_p = 1 + ((1 - 4*k)/p), where ((1 - 4*k)/p) denotes the Legendre/Kronecker symbol. Thus the local factor at odd primes p equals (1 - (1 + ((1 - 4*k)/p))/p) / (1 - 1/p).

For even k, the local factor at p = 2 is 0, and hence C(k) = 0, while for odd k the local factor at p = 2 equals 2.

In particular, the constant C(k) depends essentially on the fundamental discriminant D = 1 - 4k.

All Hardy-Littlewood constants C(k) were computed using the PARI/GP method described by Belabas and Cohen (2020), which allows high-precision numerical evaluation of the corresponding Euler products.

The present table was produced by independent computations.

Record detection was performed independently: C(k) was evaluated for each k in increasing order of |k|, and a new record was noted whenever C(k) exceeded all previously observed values.

The corresponding sequence for k > 0 is A331940.

Candidates for further terms after a(22) that still need confirmation are 752293, 1354363. - Hugo Pfoertner, Apr 02 2026

Links

Table of n, a(n) for n=1..22.

Karim Belabas and Henri Cohen, Computation of the Hardy-Littlewood constant for quadratic polynomials, PARI/GP script, 2020.

Keith Conrad, Hardy-Littlewood Constants, (2003).

Example

For k = -1, the discriminant is D = 5. The quadratic x^2 + x - 1 has local factor 2 at p = 2, and the local factor (1 - (1 + (D/p))/p) / (1 - 1/p) for odd primes p. Thus its Hardy-Littlewood constant is C(-1).
For k = -43, the discriminant is D = 173. The value C(-43) is larger than C(-1), so -43 is the next record.

Prog

(PARI) C(k) = if(k%2==0, 0, 2*prodeuler(p=3, 30000, (1-(1+kronecker(1-4*k, p))/p)/(1-1/p)))

Crossrefs

Cf. A331940, A331942, A331947.

Sequence in context: A144975 A087699 A054807 * A285017 A139932 A290635

Adjacent sequences: A394747 A394748 A394749 * A394751 A394753 A394754

Keyword

nonn,hard,more

Author

Nobuaki Sakai, Mar 31 2026

Extensions

a(10)-a(22) from Hugo Pfoertner, Apr 02 2026

Status

approved