Several results involving d(n!) are obtained, where d(m) denotes the number of positive divisors of m. These include estimates for d(n!)/d((n - 1)!), d(n!) - d((n - 1)!), as well as the least number K with d((n +K)!)/d(n!) # 2. 1

Let H(x) be a monic polynomial over a finite field F = GF(q). Denote by Na(n) the number of coefficients in H n which are equal to an element a ∈ F, and by G the set of elements a ∈ F × such that Na(n)> 0 for some n. we study the relationship between the numbers (Na(n))a∈G and the patterns in the base q representation of n. This enables us to prove that for “most ” n’s we have Na(n) ≈ Nb(n), a, b ∈ G. Considering the case H = x + 1, we provide new results on Pascal’s triangle modulo a prime. We also provide analogous results for the triangle of Stirling numbers of the first kind. Keywords: Linear cellular automata, Pascal’s triangle, asymptotic frequency.

Sieve methods have had a long and fruitful history. The sieve of Eratosthenes (around 3rd century B.C.) was a device to generate prime numbers. Later Legendre used it in his studies of the prime number counting function π(x). Sieve methods bloomed and became a topic of intense investigation after the pioneering work of Viggo Brun (see [Bru16],[Bru19], [Bru22]). Using his formulation of the sieve Brun proved, that the sum