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A230902 Positive numbers such that half of the set of divisors are of the form x^2 + x*y + y^2 (A003136) and half not (A034020).

%I A230902 #19 Jun 07 2024 09:24:16
%S A230902 2,5,6,8,11,14,15,17,18,23,24,26,29,32,33,35,38,41,42,45,47,51,53,54,
%T A230902 56,59,62,65,69,71,72,74,77,78,83,86,87,89,95,96,98,99,101,104,105,
%U A230902 107,113,114,119,122,123,125,126,128,131,134,135,137,141,143,146,149,152,153,155,158,159,161,162
%N A230902 Positive numbers such that half of the set of divisors are of the form x^2 + x*y + y^2 (A003136) and half not (A034020).
%e A230902 Triangle read by rows in which row n lists the divisors of n begins:
%e A230902 1(0^2+0*1+1^2);
%e A230902 1(0^2+0*1+1^2), 2;
%e A230902 1(0^2+0*1+1^2), 3(1^1+1*1+1^2);
%e A230902 1(0^2+0*1+1^2), 2, 4(0^2+0*2+2^2);
%e A230902 1(0^2+0*1+1^2), 5;
%e A230902 1(0^2+0*1+1^2), 2, 3(1^1+1*1+1^2), 6;
%e A230902 1(0^2+0*1+1^2), 7(1^1+1*2+2^2);
%e A230902 1(0^2+0*1+1^2), 2, 4(0^2+0*2+2^2), 8;
%e A230902 1(0^2+0*1+1^2), 3(1^1+1*1+1^2), 9;
%e A230902 1(0^2+0*1+1^2), 2, 5, 10;
%e A230902 1(0^2+0*1+1^2), 11;
%e A230902 1(0^2+0*1+1^2), 2, 3(1^1+1*1+1^2), 4(0^2+0*2+2^2), 6, 12(2^2+2*2+2^2);
%e A230902 1(0^2+0*1+1^2), 13(1^2+1*3+3^2);
%e A230902 1(0^2+0*1+1^2), 2, 7(1^1+1*2+2^2), 14;
%e A230902 1(0^2+0*1+1^1), 3(1^11+1*1+1^2), 5, 15,
%e A230902 i.e. a(1)=2, a(2)=5, a(3)=6, a(4)=8, a(5)=11, a(6)=14, a(7)=15.
%p A230902 isA003136 := proc(n)
%p A230902     local x,y ;
%p A230902     for x from 0 do
%p A230902         if x^2 > n then
%p A230902             return false;
%p A230902         end if;
%p A230902         for y from 0 do
%p A230902             if x^2+x*y+y^2 = n then
%p A230902                 return true;
%p A230902             elif x^2+x*y+y^2 > n then
%p A230902                 break;
%p A230902             end if;
%p A230902         end do:
%p A230902     end do:
%p A230902 end proc:
%p A230902 isA230902 := proc(n)
%p A230902     local a36,a20,d ;
%p A230902     a36 := 0 ;
%p A230902     a20 := 0 ;
%p A230902     for d in numtheory[divisors](n) do
%p A230902         if isA003136(d) then
%p A230902             a36 := a36+1 ;
%p A230902         else
%p A230902             a20 := a20+1 ;
%p A230902         end if;
%p A230902     end do:
%p A230902     simplify( a36=a20) ;
%p A230902 end proc:
%p A230902 for n from 0 to 200 do
%p A230902     if isA230902(n) then
%p A230902     printf("%d,",n);
%p A230902     end if;
%p A230902 end do: # _R. J. Mathar_, Nov 08 2013
%t A230902 A003136Q[n_] := Resolve[Exists[{x, y}, Reduce[n == x^2 + x*y + y^2, {x, y}, Integers]]];
%t A230902 okQ[n_] := With[{dd = Divisors[n]}, 2 Count[dd, _?A003136Q] == Length[dd]];
%t A230902 Select[Range[200], okQ] (* _Jean-François Alcover_, Jun 07 2024 *)
%Y A230902 Cf. A027750, A230851. Subsequence of A000037.
%K A230902 nonn
%O A230902 1,1
%A A230902 _Juri-Stepan Gerasimov_, Oct 31 2013
%E A230902 Corrected by _R. J. Mathar_, Nov 08 2013