(* Arithmetic derivative *)

ClearAll[d, twoFactorsOf];

twoFactorsOf[n_Integer?Positive] := Module[{factors = FactorInteger[n, 2], p, factor},
   If[Length[factors] == 1,
       factor = Flatten@factors;
        p = First@factor;
        factors = {factor - {0, 1}, {p, 1}};
   ];
   Return[Power@@@factors];
];

twoFactorsOf[n_Integer?Negative] := twoFactorsOf[-n] * {-1, -1};

d[0] = d[1] = 0;
d[p_Integer?PrimeQ] := 1;
d[n_Integer?Negative] := -d[-n];
d[mn_Integer] := Module[{m, n},
   {m, n} = twoFactorsOf[m n];
   Return[d[m] n + m d[n]];
];

SetAttributes[d, Listable];

(* Output *)

Partition[StringPadLeft[ToString /@ d[Range[-99, 100]], 5], UpTo[10]] // TableForm

StringJoin["d[10^", ToString@First[#], "]", If[First[#] <= 9, "  ", " "], "/ 7",
    " = ",
    ToString@Last[#]] & /@ Table[{n, d[10^n]/7}, {n, 1, 20}] // TableForm