A241482 -id:A241482 - cp's OEIS frontend

A232931 The least positive integer k such that Kronecker(D/k) = -1 where D runs through all positive fundamental discriminants (A003658).

2, 3, 5, 2, 3, 2, 7, 5, 2, 5, 2, 7, 3, 3, 2, 3, 5, 13, 2, 3, 2, 5, 7, 2, 2, 5, 3, 3, 2, 5, 2, 3, 11, 2, 3, 11, 7, 7, 2, 7, 3, 3, 2, 7, 2, 3, 11, 2, 3, 2, 5, 5, 2, 5, 2, 11, 3, 3, 5, 2, 7, 11, 2, 3, 2, 5, 7, 2, 2, 5, 3, 3, 2, 7, 3, 11, 2, 3, 7, 7, 5, 2, 5, 2, 13, 3, 3, 2, 2, 3, 2, 3, 2, 5, 5, 11, 2, 7, 5, 3, 3, 5, 2, 3, 13, 5, 2, 3, 2, 17, 2, 2, 7, 3, 3, 2, 13, 2, 5, 2, 3, 5, 7, 5, 2, 5, 2, 11, 3, 2, 5, 2, 3, 7, 2, 3, 2, 17, 5, 7, 2, 7, 2, 5, 3, 3, 7, 2, 3, 7, 5, 2, 3

Offset: 2

Steven Finch, Dec 02 2013

nonn

From Jianing Song, Jan 30 2019: (Start)
a(n) is necessarily prime. Otherwise, if a(n) is not prime, we have (D/p) = 0 or 1 for all prime divisors p of a(n), so (D/a(n)) must be 0 or 1 too, a contradiction.
a(n) is the least inert prime in the real quadratic field with discriminant D, D = A003658(n). (End)

			A003658(3) = 8, (8/3) = -1 and (8/2) = 0, so a(3) = 3.

Charles R Greathouse IV, Table of n, a(n) for n = 2..10000
S. R. Finch, Average least nonresidues, December 4, 2013. [Cached copy, with permission of the author]
Andrew Granville, R. A. Mollin and H. C. Williams, An upper bound on the least inert prime in a real quadratic field, Canad. J. Math. 52:2 (2000), pp. 369-380.
P. Pollack, The average least quadratic nonresidue modulo m and other variations on a theme of Erdős, J. Number Theory 132 (2012) 1185-1202.
Enrique Treviño, The least inert prime in a real quadratic field, Mathematics of Computation 81:279 (2012), pp. 1777-1797. See also his PANTS 2010 talk.

Cf. A003658, A092419, A241482.

nMax = 200; A003658 = Select[Range[4nMax], NumberFieldDiscriminant[Sqrt[#]] == #&]; f[d_] := For[k = 1, True, k++, If[FreeQ[{0, 1}, KroneckerSymbol[d, k]], Return[k]]]; a[n_] := f[A003658[[n]]]; Table[a[n], {n, 2, nMax}] (* Jean-François Alcover, Nov 05 2016 *)

lp(D)=forprime(p=2,,if(kronecker(D,p)<0,return(p)))
for(n=5,1e3,if(isfundamental(n),print1(lp(n)", "))) \\ Charles R Greathouse IV, Apr 23 2014

With D = A003658(n): Mollin conjectured, and Granville, Mollin, & Williams proved, that for n > 1128, a(n) <= D^0.5 / 2. Treviño proves that for n > 484, a(n) <= D^0.45. Asymptotically the best known upper bound for the exponent is less than 0.16 when D is prime and 1/4 + epsilon (for any epsilon > 0) for general D. - Charles R Greathouse IV, Apr 23 2014 (corrected by Enrique Treviño, Mar 18 2022)

a(n) = A092419(A003658(n) - floor(sqrt(A003658(n)))), n >= 2. - Jianing Song, Jan 30 2019

Name simplified by Jianing Song, Jan 30 2019

A306218 Fundamental discriminant D < 0 with the least absolute value such that the first n primes p have (D/p) >= 0, negated.

Original entry on oeis.org

4, 8, 15, 20, 24, 231, 264, 831, 920, 1364, 1364, 9044, 67044, 67044, 67044, 67044, 268719, 268719, 3604695, 4588724, 5053620, 5053620, 5053620, 5053620, 60369855, 364461096, 532735220, 715236599, 1093026360, 2710139064, 2710139064, 3356929784, 3356929784

Offset: 1

Jianing Song, Jan 29 2019

nonn

a(n) is the negated fundamental discriminant D < 0 with the least absolute value such that the first n primes either decompose or ramify in the imaginary quadratic field with discriminant D. See A241482 for the real quadratic field case.

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

Cf. A003657, A232932, A241482 (the real quadratic field case).

A045535 and A094841 are similar sequences.

a(n) = my(i=1); while(!isfundamental(-i)||sum(j=1, n, kronecker(-i,prime(j))==-1)!=0, i++); i

a(n) = A003657(k), where k is the smallest number such that A232932(k) >= prime(n+1).

a(26)-a(33) from Jinyuan Wang, Apr 06 2019

cp's OEIS Frontend

A232931 The least positive integer k such that Kronecker(D/k) = -1 where D runs through all positive fundamental discriminants (A003658).

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