80 is in the sequence since sigma(80) = 186 = 144 + 32 + 10 = psi(80) + phi(80) + tau(80).
MATHEMATICA
f1[p_, e_] := (p^(e + 1) - 1)/(p - 1);
f2[p_, e_] := (p + 1)*p^(e - 1);
f3[p_, e_] := (p - 1)*p^(e - 1);
f4[p_, e_] := e + 1;
q[k_] := Module[{f = FactorInteger[k]}, Times @@ f1 @@@ f == Times @@ f2 @@@ f + Times @@ f3 @@@ f + Times @@ f4 @@@ f]; Select[Range[2, 6*10^5], q] (* Amiram Eldar, Oct 14 2025 *)
