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

1,1

Comments

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

Links

Table of n, a(n) for n=1..58.

Example

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

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

Crossrefs

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

Sequence in context: A073712 A157200 A255940 * A354361 A018511 A345139

Adjacent sequences: A167412 A167413 A167414 * A167416 A167417 A167418

Keyword

easy,nonn

Author

Arnaud Vernier, Nov 03 2009

Extensions

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

Status

approved