Prime signature

(*start*)
Clear[A, b, nn, n, x];
nn = 500;
b = Inverse[
   A = Table[
      Table[If[Mod[n, k] == 0, 1 + x, 0], {k, 1, nn}], {n, 1, nn}] -
     IdentityMatrix[nn]*x][[All, 1]]
TableForm[Table[{n, b[[n]]}, {n, 1, 20}]]
S = DeleteDuplicates[
   Table[Flatten[Position[b, b[[n]]]], {n, 2, Length[b]}]];
"Polynomial first:"
Do[Print[{b[[S[[n]]]][[1]], S[[n]]}], {n, 1, Length[S]}]
Clear[p, q, r, s, t];
primesignature = {p, p^2, p*q, p^3, p^2*q, p^4, p^3*q, p*q*r, p^5,
   p^2*q^2, p^4*q, p^2*q*r, p^6, p^3*q^2, p^5*q, p^3*q*r, p^7,
   p^4*q^2, p^2*q^2*r, p^6*q, p*q*r*s};
TableForm[
 Table[{primesignature[[n]], b[[S[[n]]]][[1]]}, {n, 1,
   Length[primesignature]}]]
(*end*)