A241482 Least fundamental discriminant D > 1 such that the first n primes p have (D/p) >= 0.

8, 12, 24, 60, 60, 364, 984, 1596, 1596, 1596, 3705, 58444, 84396, 164620, 172236, 369105, 369105, 731676, 731676, 3442296, 3442296, 32169916, 32169916, 47973864, 47973864, 47973864, 313114620, 313114620, 313114620, 313114620, 13461106065, 27765196680, 40527839121, 55213498824, 55213498824, 381031123720

Offset: 1

Author

Charles R Greathouse IV, Apr 23 2014

Keywords

• nonn

Comments

By the Chinese Remainder Theorem and Prime Number Theorem in arithmetic progressions, this sequence is infinite.

a(n) is the least fundamental discriminant D > 1 such that the first n primes either decompose or ramify in the real quadratic field with discriminant D. See A306218 for the imaginary quadratic field case. - Jianing Song, Feb 14 2019

Examples

			(364/2) = 0, (364/3) = 1, (364/5) = 1, (364/7) = 0, (364/11) = 1, (364/13) = 0, so 3, 5, 11 decompose in Q[sqrt(91)] and 2, 7, 13 ramify in Q[sqrt(-231)]. For other fundamental discriminants 1 < D < 364, at least one of 2, 3, 5, 7, 11, 13 is inert in the imaginary quadratic field with discriminant D, so a(6) = 364. - _Jianing Song_, Feb 14 2019
		

Crossrefs

Cf. A003658, A232931, A306218 (the imaginary quadratic field case).

A002189 and A094847 are similar sequences.

Programs

PARI

a(n) = my(i=2); while(!isfundamental(i)||sum(j=1, n, kronecker(i,prime(j))==-1)!=0, i++); i \\ Jianing Song, Feb 14 2019

Formula

a(n) > prime(n)^(4*sqrt(e) + o(1)). - Charles R Greathouse IV, Apr 23 2014

a(n) = A003658(k), where k is the smallest number such that A232931(k) >= prime(n+1). - Jianing Song, Feb 15 2019

Extensions

a(36) from Charles R Greathouse IV, Apr 24 2014