

A232931


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


10



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



