Exponents of ec primes ending with digit 7

Question

A conjecture about numbers of the form $10^{m}(2^{k}−1)+2^{k-1}−1$, where $m$ is the number of decimal digits of $ 2^{k-1}$. In this question ec-numbers are introduced, obttained by the concatenation of two consecutive Mersenne numbers (12763=ec(7) for example). The values of $k$ for which $ec(k)$ is prime are: $[2, 3, 4, 7, 8, 12, 19, 22, 36, 46, 51, 67, 79, 215, 359, 394, 451, 1323, 2131, 3336, 3371, 6231, 19179, 39699, 51456, 56238, 69660, 75894, 79798, 92020, 174968, 176006, 181015, 285019, 331259, 360787, 366770]$ $7, 67, 360787$ are all the $k's$ leading to a prime ending with digit $7$. $7,67,360787$ are all congruent to $1$ $\pmod 6$. My question is: does a k exist such that k ends with digit $7$, $ec(k)$ is prime and $k$ is NOT congruent to 1 $\pmod 6$? Or that can be ruled out? In other words, if $k$ is congruent to 7 mod 10 and $ec(k)$ is prime, then necessarly $k$ must be congruent to 1 mod 6?