OEIS Diophantine relation A366346

f = y^2 + y - (x^3 - x^2);
nn = 12;
constant = 0; (* or 0 *)
TableForm[
 BB = Table[
   Table[-Prime[n]*(Prime[n] - 1)*constant*1 +
     Sum[Sum[If[GCD[f, Prime[n]] == g, 1, 0], {x, 1, Prime[n]}], {y,
       1, Prime[n]}], {n, 1, nn}], {g, 1, 4*nn}]]
Total[BB]
TableForm[
 CC = Sign[
   Table[Table[
     Sum[Sum[If[GCD[f, Prime[n]] == g, 1, 0], {x, 1, Prime[n]}], {y,
       1, Prime[n]}], {n, 1, nn}], {g, 1, 4*nn}]]]
Total[CC]
Monitor[TableForm[
  AA = Table[
     Table[-Prime[n]*(Prime[n] - 1)*constant +
       Sum[Sum[If[GCD[f, Prime[n]] == g, 1, 0], {x, 1, Prime[n]}], {y,
          1, Prime[n]}], {n, 1, nn}], {g, 1, 4*nn}]*
    Sign[Table[
      Table[Sum[
        Sum[If[GCD[f, Prime[n]] == g, 1, 0], {x, 1, Prime[n]}], {y, 1,
          Prime[n]}], {n, 1, nn}], {g, 1, 4*nn}]]], g]
Total[AA]
Sum[Table[AA[[n, k]], {n, 2, Length[AA]}], {k, 1, nn}]