Prime factors of binomial coefficients and related problems

Abstract

1 . Sequences of positive integers with the consecutive integer property. Consider a sequence of k positive integers {ai} = a,, . . ., a k with the following properties : (i) ai < k for all i, (u) (the consecutive integer property) There is an n such that ai is the quotient when n + i is cleared of all its prime factors greater than k . Or, for each prime p k, the pattern of that prime and its powers in the a's is the same as the pattern of that prime and its powers in some sequence of k consecutive integers . Notice that the sequence l, 4, 3, 2 satisfies the above properties since the consecutive integers 19, 20, 21, 22 factor into 1 • 19, 4 . 5, 3 . 7 2 . 11 . The sequences 2, 3, 1 and 2, 3, 2 satisfy (i) but not (u) since if n + 1 is twice an odd number, then 4 divides n + 3 . The sequence l, 6, 1 satisfies (u) (with n + 1 = 5) but not (i) .