A306220 -id:A306220 - cp's OEIS frontend

A232932 The least positive integer k such that Kronecker(D/k) = -1 where D runs through all negative fundamental discriminants (-A003657).

2, 3, 3, 5, 2, 7, 2, 11, 5, 13, 3, 2, 7, 3, 2, 5, 2, 3, 3, 11, 2, 2, 5, 7, 3, 2, 13, 5, 3, 2, 7, 3, 11, 2, 11, 2, 7, 11, 7, 2, 3, 2, 5, 3, 2, 5, 3, 3, 5, 2, 11, 2, 13, 5, 5, 2, 5, 3, 2, 7, 2, 3, 2, 2, 5, 13, 2, 3, 2, 5, 17, 3, 2, 7, 3, 3, 5, 2, 13, 2, 7, 5, 19, 2, 3, 11, 3, 2, 5, 2, 3, 3, 7, 2, 5, 2, 5, 11, 5, 3, 2, 5, 3, 2, 11, 2, 3, 7, 2, 2, 11, 7, 3, 2, 5, 3, 2, 5, 3, 3, 2, 11, 2, 19, 5, 5, 2, 3, 2, 17, 3, 2, 7, 2, 3, 3, 13, 2, 5, 2, 5, 11, 7, 3, 2, 7, 3, 13, 2, 3, 5, 2, 2

Offset: 1

Steven Finch, Dec 02 2013

nonn

From Jianing Song, Feb 14 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 imaginary quadratic field with discriminant D, D = -A003657(n). (End)

			A003657(4) = 8, (-8/5) = -1, (-8/3) = 1 and (-8/2) = 0, so a(4) = 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000
S. R. Finch, Average least nonresidues, December 4, 2013. [Cached copy, with permission of the author]
P. Pollack, The average least quadratic nonresidue modulo m and other variations on a theme of Erdős, J. Number Theory 132 (2012) 1185-1202.

Cf. A003657, A306220.

nMax = 200; FundamentalDiscriminantQ[n_] := n != 1 && (Mod[n, 4] == 1 || ! Unequal[Mod[n, 16], 8, 12]) && SquareFreeQ[n/2^IntegerExponent[n, 2]]; discrims = Select[-Range[4 nMax], FundamentalDiscriminantQ]; f[d_] := For[k = 1, True, k++, If[FreeQ[{0, 1}, KroneckerSymbol[d, k]], Return[k] ] ]; a[n_] := f[discrims[[n]]]; Table[a[n], {n, 1, nMax}] (* Jean-François Alcover, Nov 05 2016, after Robert G. Wilson v *)

a(n) = A306220(A003657(n)). - Jianing Song, Feb 14 2019

Name simplified by Jianing Song, Feb 14 2019

A306224 a(n) is the smallest prime p such that -n is not a square mod p.

Original entry on oeis.org

3, 5, 5, 3, 11, 13, 3, 5, 7, 3, 7, 5, 3, 11, 7, 3, 5, 5, 3, 11, 13, 3, 5, 13, 3, 11, 5, 3, 7, 7, 3, 5, 5, 3, 19, 7, 3, 5, 7, 3, 13, 5, 3, 7, 11, 3, 5, 5, 3, 7, 7, 3, 5, 13, 3, 11, 5, 3, 11, 7, 3, 5, 5, 3, 7, 19, 3, 5, 11, 3, 7, 5, 3, 7, 11, 3, 5, 5, 3, 11, 7, 3, 5, 13, 3, 7, 5

Offset: 1

Jianing Song, Jan 30 2019

nonn

a(n) is the smallest odd prime p such that Kronecker(-n,p) = -1.
A companion sequence to A144294.
Conjecture: lim sup log(a(n))/log(n) = 0.

Cf. A144294.

See A306220 for another version.

a(n) = forprime(p=3, , if(kronecker(-n, p)<0, return(p)))

a(n) = 3 if and only if n == 1 (mod 3).

a(n) = 5 if and only if n == 2, 3, 8, 12 (mod 15).

cp's OEIS Frontend

A232932 The least positive integer k such that Kronecker(D/k) = -1 where D runs through all negative fundamental discriminants (-A003657).

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