### Program for Sage

The ec(k) numbers are so defined:
ec(k)=(2^k-1)*10^d+2^(k-1)-1, where d is the number of decimal digits of 2^(k-1)-1.
Examples of these numbers are: 31, 157, 3115, 40952047,...
I found that up to k=565.000 there is no prime of the form (2^k-1)*10^d+2^(k-1)-1 which is congruent to 6 mod 7, so I conjectured that there is no prime of this form congruent to 6 mod 7. Has somebody a program for Sage for checking this conjecture further?