criteria to verify log-convexity of sequences is presented. Iterating this criteria produces infinitely log-convex sequences. As an application, several classical examples of sequences arising in Combinatorics and Special Functions are presented. The paper concludes with a conjecture regarding coefficients of chromatic polynomials.

We discuss the VC-dimension of a class of multiples of integers and primes (equivalently indicator functions) and demonstrate connections to prime counting functions. Additionally, we prove limit theorems for the behavior of an empirical risk minimization rule as well as the weights assigned to the output hypothesis in AdaBoost for these “prime-identifying” indicator functions, when we sample \( mn \) i.i.d. points uniformly from the integers \(\{2, \ldots, n\}\).

Integer compositions and related counting problems are a rich and ubiquitous topic in enumerative combinatorics. In this paper we explore the definition of symmetric and asymmetric peaks and valleys over compositions. In particular, we compute an explicit formula for the generating function for the number of integer compositions according to the number of parts, symmetric, and asymmetric peaks and valleys.

In this paper we show some identities come from the \( q \)-identities of Euler, Jacobi, Gauss, and Rogers-Ramanujan. Some of these identities relate the function sum of divisors of a positive integer \( n \) and the number of integer partitions of \( n \). One of the most intriguing results found here is given by the next equation, for \( n > 0 \),
\[
\sum_{l=1}^n \frac{1}{l!} \sum_{w_1+w_2+\cdots+w_l \in C(n)} \frac{\sigma_1(w_1) \sigma_1(w_2) \cdots \sigma_1(w_l)}{w_1 w_2 \cdots w_l} = p_1(n),
\]
where \( \sigma_1(n) \) is the sum of all positive divisors of \( n \), \( p_1(n) \) is the number of integer partitions of \( n \), and \( C(n) \) is the set of integer compositions of \( n \). In the last section, we show seven applications, one of them is a series expansion for
\[
\frac{(q^{a_1};q^{b_1})_\infty (q^{a_2};q^{b_2})_\infty \cdots (q^{a_k};q^{b_k})_\infty}
{(q^{c_1};q^{d_1})_\infty (q^{c_2};q^{d_2})_\infty \cdots (q^{c_r};q^{d_r})_\infty},
\]
where \( a_1, \ldots, a_k, b_1, \ldots, b_k, c_1, \ldots, c_r, d_1, \ldots, d_r \) are positive integers, and \( |q| < 1 \).

Between Bernoulli/Euler polynomials and Pell/Lucas polynomials, convolution sums are evaluated in closed form via the generating function method. Several interesting identities involving Fibonacci and Lucas numbers are shown as consequences including those due to Byrd \( (1975) \) and Frontczak \( (2020) \).

The notion of length spectrum for natural numbers was introduced by Pong in \([5]\). In this article, we answer the question of how often one can recover a random number from its length spectrum. We also include a quick deduction of a result of LeVeque in \([4]\) on the average order of the size of length spectra.