A333669 The smallest nontrivial quadratic residue modulo n.

4, 3, 2, 4, 4, 4, 3, 4, 3, 2, 4, 4, 2, 4, 4, 4, 4, 3, 2, 4, 4, 3, 4, 4, 4, 4, 2, 4, 3, 2, 4, 4, 3, 4, 3, 4, 2, 4, 4, 4, 4, 2, 2, 4, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 4, 3, 2, 4, 4, 4, 3, 4, 4, 3, 4, 2, 4, 2, 3, 4, 4, 4, 3, 2, 4, 4, 2, 3, 4, 4, 4, 4, 4

Offset

5,1

Comments

The values are 2, 3 and 4. If 2 is a square modulo n (see A057126) the value is 2. Otherwise, if 3 is a square modulo n (see A057125) the value is 3. If neither 2 or 3 are a square modulo n the value is 4.

Dedicated to Urs Meyer at the occasion of his 60th birthday.

Links

Robert Israel, Table of n, a(n) for n = 5..10000

Example

The squares modulo 5 are 1 and 4, therefore a(5) = 4.
Modulo 6 the squares are 1, 3 and 4 which makes a(6) = 3.
a(7) = 2 since 2 == 3^2 (mod 7).

Maple

f:= proc(n) uses numtheory; if quadres(2, n)=1 then 2 elif quadres(3, n)=1 then 3 else 4 fi end proc:
map(f, [$5..100]); # Robert Israel, Sep 15 2020

Mathematica

qrQ[m_, n_] := Module[{k}, Reduce[Mod[m-k^2, n]==0, k, Integers] =!= False];
a[n_] := If[qrQ[2, n], 2, If[qrQ[3, n], 3, 4]];
a /@ Range[5, 100] (* Jean-François Alcover, Oct 25 2020 *)

Prog

(PARI) a(n) = if(issquare(Mod(2, n)), 2, issquare(Mod(3, n)), 3, 4)

Crossrefs

Cf. A057126 for the n where the value is 2 and A057125 for the n where the value is 3 if n was not in A057126.

Sequence in context: A304240 A244951 A110631 * A159846 A377512 A071890

Adjacent sequences: A333666 A333667 A333668 * A333670 A333671 A333672

Keyword

nonn,easy

Author

Peter Schorn, May 07 2020

Status

approved