

A232931


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


9



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
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OFFSET

2,1


COMMENTS

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)


LINKS

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. 369380.
P. Pollack, The average least quadratic nonresidue modulo m and other variations on a theme of Erdős, J. Number Theory 132 (2012) 11851202.
Enrique Treviño, The least inert prime in a real quadratic field, Mathematics of Computation 81:279 (2012), pp. 17771797. See also his PANTS 2010 talk.


FORMULA

With D = A003658(n): Mollin conjectured, and Granville, Mollin, & Williams prove, that for n > 1128, a(n) <= D^0.5 / 2. Treviño proves that for n > 484, a(n) <= D^0.45. Asymptotically the exponent is less than 0.16.  Charles R Greathouse IV, Apr 23 2014
a(n) = A092419(A003658(n)  floor(sqrt(A003658(n))), n >= 2.  Jianing Song, Jan 30 2019


EXAMPLE

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


MATHEMATICA

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}] (* JeanFrançois Alcover, Nov 05 2016 *)


PROG

(PARI) 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


CROSSREFS

Cf. A003658, A092419, A241482.
Sequence in context: A133906 A317358 A133907 * A060084 A265668 A273087
Adjacent sequences: A232928 A232929 A232930 * A232932 A232933 A232934


KEYWORD

nonn


AUTHOR

Steven Finch, Dec 02 2013


EXTENSIONS

Name simplified by Jianing Song, Jan 30 2019


STATUS

approved



