A394750 - OEIS

%I #53 Apr 18 2026 16:49:44

%S 1,43,73,109,283,619,2293,6163,7759,23929,42739,56269,82009,90073,

%T 153409,169933,211999,249439,321889,328063,349513,501229

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

%C 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).

%C 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).

%C 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.

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

%C 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.

%C The present table was produced by independent computations.

%C 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.

%C The corresponding sequence for k > 0 is A331940.

%C Candidates for further terms after a(22) that still need confirmation are 752293, 1354363. - _Hugo Pfoertner_, Apr 02 2026

%H Karim Belabas and Henri Cohen, <a href="/A221712/a221712.gp.txt">Computation of the Hardy-Littlewood constant for quadratic polynomials</a>, PARI/GP script, 2020.

%H Keith Conrad, <a href="https://kconrad.math.uconn.edu/articles/hlconst.pdf">Hardy-Littlewood Constants</a>, (2003).

%e 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).

%e For k = -43, the discriminant is D = 173. The value C(-43) is larger than C(-1), so -43 is the next record.

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

%Y Cf. A331940, A331942, A331947.

%K nonn,hard,more

%O 1,2

%A _Nobuaki Sakai_, Mar 31 2026

%E a(10)-a(22) from _Hugo Pfoertner_, Apr 02 2026