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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
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. 369-380.
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.
FORMULA
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
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}] (* Jean-François 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
KEYWORD
nonn
AUTHOR
Steven Finch, Dec 02 2013
EXTENSIONS
Name simplified by Jianing Song, Jan 30 2019
STATUS
approved