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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).
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.
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
KEYWORD
easy,nonn
AUTHOR
Arnaud Vernier, Nov 03 2009
EXTENSIONS
Name corrected by R. J. Mathar, Jun 16 2019 and Don Reble
STATUS
approved