cp's OEIS Frontend

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).

%I A167415 #23 May 17 2024 09:56:06
%S A167415 2,3,6,7,13,14,17,21,23,26,34,37,39,42,43,46,47,51,53,67,69,73,74,78,
%T A167415 83,86,91,94,97,102,103,106,107,111,113,119,127,129,134,137,138,141,
%U A167415 146,157,159,161,163,166,167,173,182,193,194,197,201,206,214,219
%N 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).
%C A167415 Prime numbers of this sequence are congruent to {2,3} modulo 5.
%e A167415 The only solution of the equation x^2 + y^2 + 3*x*y = 0 in Z/2Z is (0,0).
%e A167415 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
%p A167415 isA167415 := proc(n)
%p A167415     local x,y ;
%p A167415     for x from 0 to n-1 do
%p A167415         for y from x to n-1 do
%p A167415             if modp(x^2+y^2+3*x*y,n) = 0 and (x <> 0 or y <> 0) then
%p A167415                 return false;
%p A167415             end if;
%p A167415         end do:
%p A167415     end do:
%p A167415     true ;
%p A167415 end proc:
%p A167415 for n from 2 to 300 do
%p A167415     if isA167415(n) then
%p A167415         printf("%d,",n) ;
%p A167415     end if;
%p A167415 end do: # _R. J. Mathar_, Jun 16 2019
%Y A167415 Cf. A031363 (x^2 + y^2 + 3xy).
%K A167415 easy,nonn
%O A167415 1,1
%A A167415 _Arnaud Vernier_, Nov 03 2009
%E A167415 Name corrected by _R. J. Mathar_, Jun 16 2019 and _Don Reble_