OFFSET
1,1
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
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
EXAMPLE
(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
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Charles R Greathouse IV, Apr 23 2014
EXTENSIONS
a(36) from Charles R Greathouse IV, Apr 24 2014
STATUS
approved