Log operator Euler products

(*program start*)Clear[nn, logarithm, LOGPRODUCT, LOGOPERATOR, n, k]
nn = 90; logarithm = 1; LOGPRODUCT =
 Table[Table[
   If[n/k == logarithm, n/k, If[n == k, 1, 0]], {k, 1, nn}], {n, 1,
   nn}]; Monitor[Do[logarithm = If[PrimeQ[i] == True, i, 0];
  LOGOPERATOR =
   Table[Table[
     If[n/k == logarithm, -n/k, If[n == k, 1, 0]], {k, 1, nn}], {n, 1,
      nn}]; LOGPRODUCT = LOGPRODUCT.Inverse[LOGOPERATOR];, {i, 2,
   nn}], i]; LOGPRODUCT[[All, 1]]
(*program end*)


(*program start*)Clear[nn, logarithm, LOGPRODUCT, LOGi, n, k]
nn = 90; logarithm = 1; LOGPRODUCT =
 Table[Table[
   If[n/k == logarithm, n/k, If[n == k, 1, 0]], {k, 1, nn}], {n, 1,
   nn}]; Monitor[Do[logarithm = If[PrimeQ[i] == True, i, 0];
  LOGi = Table[
    Table[If[n/k == logarithm, -n/k, If[n == k, 1, 0]], {k, 1,
      nn}], {n, 1, nn}];
  LOGPRODUCT = LOGPRODUCT.LOGi;, {i, 2, nn}], i]; LOGPRODUCT[[All, 1]]
(*program end*)



(*program start*)Clear[nn, logarithm, LOGPRODUCT, LOGOPERATOR, n, k]
nn = 90;
logarithm = 1; LOGPRODUCT =
 Table[Table[
   If[n/k == logarithm, n/k, If[n == k, 1, 0]], {k, 1, nn}], {n, 1,
   nn}];
Monitor[Do[logarithm = If[PrimeQ[i] == True, i, 0];
  LOGOPERATOR =
   Table[Table[
     If[n/k == logarithm, -n/k, If[n == k, 1, 0]], {k, 1, nn}], {n, 1,
      nn}];
  LOGPRODUCT =
   LOGPRODUCT.Inverse[LOGOPERATOR].Inverse[LOGOPERATOR];, {i, 2,
   nn}], i]; LOGPRODUCT[[All, 1]]/Range[nn]
(*program end*)