Let $k\ge 3$ be an intger. The object is to find the smallest number $n$ above $10^k$ which is the product of two distinct primes, each greater than $11$ and containing at most $2$ distinct decimal digits. The numbers for $3$ upto $11$ are :
? for(s=3,11,n=10^s;gef=0;while(gef==0,n=n+1;w=factor(n);if(Set(component(w,2))=
=[1],if(length(component(w,1))==2,if(component(w,1)[1]>11,if(length(Set(digits(c
omponent(w,1)[1])))<=2,if(length(Set(digits(component(w,1)[2])))<=2,gef=1))))));
print(s," ",n," ",w))
3 1003 [17, 1; 59, 1]
4 10019 [43, 1; 233, 1]
5 100091 [101, 1; 991, 1]
6 1002013 [31, 1; 32323, 1]
7 10002373 [449, 1; 22277, 1]
8 100005887 [8999, 1; 11113, 1]
9 1000047757 [11113, 1; 89989, 1]
10 10000252373 [449, 1; 22272277, 1]
11 100004653457 [11113, 1; 8998889, 1]
Which numbers come next ? In particular, what is the number for $k=12$ ?