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).

2, 5, 6, 8, 11, 14, 15, 17, 18, 23, 24, 26, 29, 32, 33, 35, 38, 41, 42, 45, 47, 51, 53, 54, 56, 59, 62, 65, 69, 71, 72, 74, 77, 78, 83, 86, 87, 89, 95, 96, 98, 99, 101, 104, 105, 107, 113, 114, 119, 122, 123, 125, 126, 128, 131, 134, 135, 137, 141, 143, 146, 149, 152, 153, 155, 158, 159, 161, 162

Offset: 1

Author

Juri-Stepan Gerasimov, Oct 31 2013

Keywords

• nonn

Examples

			Triangle read by rows in which row n lists the divisors of n begins:
1(0^2+0*1+1^2);
1(0^2+0*1+1^2), 2;
1(0^2+0*1+1^2), 3(1^1+1*1+1^2);
1(0^2+0*1+1^2), 2, 4(0^2+0*2+2^2);
1(0^2+0*1+1^2), 5;
1(0^2+0*1+1^2), 2, 3(1^1+1*1+1^2), 6;
1(0^2+0*1+1^2), 7(1^1+1*2+2^2);
1(0^2+0*1+1^2), 2, 4(0^2+0*2+2^2), 8;
1(0^2+0*1+1^2), 3(1^1+1*1+1^2), 9;
1(0^2+0*1+1^2), 2, 5, 10;
1(0^2+0*1+1^2), 11;
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);
1(0^2+0*1+1^2), 13(1^2+1*3+3^2);
1(0^2+0*1+1^2), 2, 7(1^1+1*2+2^2), 14;
1(0^2+0*1+1^1), 3(1^11+1*1+1^2), 5, 15,
i.e. a(1)=2, a(2)=5, a(3)=6, a(4)=8, a(5)=11, a(6)=14, a(7)=15.
		

Crossrefs

Cf. A027750, A230851. Subsequence of A000037.

Programs

Maple

isA003136 := proc(n)
    local x,y ;
    for x from 0 do
        if x^2 > n then
            return false;
        end if;
        for y from 0 do
            if x^2+x*y+y^2 = n then
                return true;
            elif x^2+x*y+y^2 > n then
                break;
            end if;
        end do:
    end do:
end proc:
isA230902 := proc(n)
    local a36,a20,d ;
    a36 := 0 ;
    a20 := 0 ;
    for d in numtheory[divisors](n) do
        if isA003136(d) then
            a36 := a36+1 ;
        else
            a20 := a20+1 ;
        end if;
    end do:
    simplify( a36=a20) ;
end proc:
for n from 0 to 200 do
    if isA230902(n) then
    printf("%d,",n);
    end if;
end do: # R. J. Mathar, Nov 08 2013

Mathematica

A003136Q[n_] := Resolve[Exists[{x, y}, Reduce[n == x^2 + x*y + y^2, {x, y}, Integers]]];
okQ[n_] := With[{dd = Divisors[n]}, 2 Count[dd, _?A003136Q] == Length[dd]];
Select[Range[200], okQ] (* Jean-François Alcover, Jun 07 2024 *)

Extensions

Corrected by R. J. Mathar, Nov 08 2013