Generating \(d\)-composite sandwich numbers (Q2829180)

Let \(d\in\{1,2,\dots, 9\}\), \(k\) a positive integer with \(\text{gcd}(k,10k)=1\) and \(s_n(k,d)\) the decimal number \(k\undersetbrace n\to{dd\cdots d}k\). \(k\) is called a \(d\)-composite sandwich number, if \(s_n(k,d)\) is composite for all \(n\geq 1\), and it is called trivial, if they are divided by the same prime. The authors prove, that there exist infinitely many primes \(k\), which are simultaneously trivial \(d\)-composite sandwich numbers for all digits \(d\). They establish more results concerning which types of integers can be \(d\)-composite sandwich numbers, some using computer computations.