A167415 Positive integers k such that there is no solution of the equation x^2 + y^2 + 3*x*y = 0 in Z/nZ except for the trivial one (0,0).

2, 3, 6, 7, 13, 14, 17, 21, 23, 26, 34, 37, 39, 42, 43, 46, 47, 51, 53, 67, 69, 73, 74, 78, 83, 86, 91, 94, 97, 102, 103, 106, 107, 111, 113, 119, 127, 129, 134, 137, 138, 141, 146, 157, 159, 161, 163, 166, 167, 173, 182, 193, 194, 197, 201, 206, 214, 219

Offset: 1

Author

Arnaud Vernier, Nov 03 2009

Keywords

• easy
• nonn

Comments

Prime numbers of this sequence are congruent to {2,3} modulo 5.

Examples

			The only solution of the equation x^2 + y^2 + 3*x*y = 0 in Z/2Z is (0,0).
4 is not in the sequence because 0^2 + 2^2 + 3*2*0 = 4 == 0 (mod 4). 5 is not in the sequence because 1^2 + 1^2 + 3*1*1 = 5 == 0 (mod 5). 10 is not in the sequence because 2^2 + 2^2 + 3*2*2 = 20 == 0 (mod 10). - _R. J. Mathar_, Jun 16 2019
		

Crossrefs

Cf. A031363 (x^2 + y^2 + 3xy).

Programs

Maple

isA167415 := proc(n)
    local x,y ;
    for x from 0 to n-1 do
        for y from x to n-1 do
            if modp(x^2+y^2+3*x*y,n) = 0 and (x <> 0 or y <> 0) then
                return false;
            end if;
        end do:
    end do:
    true ;
end proc:
for n from 2 to 300 do
    if isA167415(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jun 16 2019

Extensions

Name corrected by R. J. Mathar, Jun 16 2019 and Don Reble