Your code has several problems. Please revise it and check the result of every step to debug it. Examples if the problems:

• You define f[ ] with several argument however, you call it with only one argument. In your case you need to define it only with f[n_]:=.....
• In the upper code part, n is a list of numbers. But at the button, you pass only one number to n when you call f[n] inside Table[ ], that is why g1 and g2 gets only one number and as a result Riffle gives errors.

Hope this helps.

Then I guess you should have mentioned the errors you are aware of to ease our members' task to help.
This part of your code works as intended (I removed the extra arguments)

ClearAll@f
f[n_] := Module[{z,z1},
  z= (n-1)^2*(n+1)^3;
g=Mod[z,3];
g2=Mod[z,5];
g3=Mod[z,7];
g4=Mod[z,11];
z1=(n-1)^2*(n+5)^3;
g1=Mod[z1,3];
g5=Mod[z1,11];
q=Riffle[g2,g3];
q1=Partition[q,2];
p=Riffle[g,g1];
q2=Partition[p,2];
r=Riffle[q1,q2];
h=Partition[r,2];
q3=Riffle[g4,g5];
q4=Partition[q3,2];
q5=Riffle[h,q4];
q6=Partition[q5,2]]

Your problem lies in Table[ ]. It is not clear what you are trying to do in this part

Table[If[MemberQ[q6, f[n]], {n, f[n]}, Nothing], {n, 1, 9961}]
Table[If[MemberQ[q6, f[n]], {n,(n-1)^2*(n+1)^3, f[n]}, Nothing], {n, 1, 9961}]
Table[If[MemberQ[q6, f[n]], {n,PrimeQ[(((n-1)^2*(n+1)^3)/7)+g3], f[n]}, Nothing], {n, 1, 9961}]

Please explain clearly the purpose of this part so that our community can help you.