A306220 a(n) is the smallest prime p such that Kronecker(-n,p) = -1.

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

Offset: 1

Author

Jianing Song, Jan 29 2019

Keywords

• nonn

Comments

A companion sequence to A092419.

Conjecture: lim sup log(a(n))/log(n) = 0. For example, it seems that log(a(n))/log(n) < 0.5 for all n > 1364.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A092419.

See A306224 for another version.

Programs

Maple

# This requires Maple 2016 or later
f:= proc(n) local p;
  p:= 2;
  while NumberTheory:-KroneckerSymbol(-n,p) <> -1 do p:= nextprime(p) od:
  p
end proc:
map(f, [$1..100]); # Robert Israel, Feb 17 2019

Mathematica

a[n_] := For[p = 2, True, p = NextPrime[p], If[KroneckerSymbol[-n, p] == -1, Return[p]]];
Array[a, 100] (* Jean-François Alcover, Jun 18 2020 *)

PARI

a(n) = forprime(p=2, , if(kronecker(-n, p)<0, return(p)))

Formula

a(n) = 2 if and only if n == 3, 5 (mod 8). See A047621.

a(n) = 3 if and only if n == 1, 4, 7, 10, 16, 22 (mod 24).