# A 10 INTEGERS 15 A ( 2015 ) GENERATING d - COMPOSITE SANDWICH NUMBERS

Let d 2 D = {1, . . . , 9}, and let k be a positive integer with gcd(k, 10d) = 1. Define a sequence {sn(k, d)}n=1 by sn(k, d) := k dd . . . d | {z } n k. We say k is a d-composite sandwich number if sn(k, d) is composite for all n 1. For a d-composite sandwich number k, we say k is trivial if sn(k, d) is divisible by the same prime for all n 1, and nontrivial otherwise. In this paper, we develop a simple criterion to determine when a d-composite sandwich number is nontrivial, and we use it to establish many results concerning which types of integers can be d-composite sandwich numbers. For example, we prove that there exist infinitely many primes that are simultaneously trivial d-composite sandwich numbers for all d 2 D. We also show that there exist infinitely many positive integers that are simultaneously nontrivial d-composite sandwich numbers for all d 2 D, where D ⇢ D with |D| = 4 and D 6= {3, 6, 7, 9}.