A354620 Numbers k such that whenever the sum of three squares is divisible by k, at least two of the squares are congruent mod k.

1, 2, 3, 4, 6, 8, 9, 11, 12, 16, 24, 32, 33, 36, 44, 48, 64, 96, 128, 132, 144, 176, 192, 256, 384, 512, 528, 576, 704, 768, 1024, 1536, 2048, 2112, 2304, 2816, 3072, 4096, 6144, 8192, 8448, 9216, 11264, 12288, 16384, 24576, 32768, 33792, 36864, 45056, 49152, 65536, 98304, 131072, 135168, 147456

Offset

1,2

Comments

If k is in the sequence and k = a*b with a and b coprime, then a and b are in the sequence. The converse is not true (thus 2 and 9 are in the sequence but 18 is not).

Links

Table of n, a(n) for n=1..56.

Mathematics Stack Exchange, Sum of squares and quadratic residues

Example

5 is not in the sequence because 0^2 + 1^2 + 2^2 = 5 is divisible by 5, with 0^2 = 0, 1^2 = 1 and 2^2 = 4 all distinct mod 5.
a(5) = 6 is in the sequence because the solutions to a^2 + b^2 + c^2 == 0 (mod 6) have (up to permutation) [a^2, b^2, c^2] == [0, 0, 0], [0, 3, 3], [1, 1, 4],  or [4, 4, 4], all of which have at least two congruent mod 6.

Maple

filter:= proc(n) local R, i, j, nR, a, b, c;
  R:= {seq(i^2 mod n, i=0..n-1)}:
  nR:= nops(R);
  for i from 2 to nR do
    a:= R[i];
    for j from 1 to i-1 do
       b:= R[j];
       c:= -a-b mod n;
       if member(c, R) and c <> a and c <> b then return false fi;
  od od;
  true
end proc:
select(filter, [$1..100000]);

Mathematica

filter[n_] := Module[{R, i, j, nR, a, b, c}, R = Union@ Table[Mod[i^2, n], {i, 0, n-1}]; nR = Length[R]; For[i = 2, i <= nR, i++, a = R[[i]]; For[j = 1, j <= i-1, j++, b = R[[j]]; c = Mod[-a-b, n]; If[MemberQ[R, c] && c != a && c != b, Return[False]]]]; True];
Select[Range[10000], filter] (* Jean-François Alcover, May 12 2023, after Robert Israel *)

Crossrefs

Sequence in context: A282697 A002479 A010458 * A218980 A084829 A018649

Adjacent sequences: A354617 A354618 A354619 * A354621 A354622 A354623

Keyword

nonn

Author

Robert Israel, Jul 10 2022

Status

approved