Counting appearances of integers in sets of arithmetic progressions
Florian Pausinger
TL;DR
This paper establishes a direct counting interpretation for the determinant-defined sequence $A067549$: for the first $k$ primes and their product $P_k$, $a_k$ equals the number of integers in $[1,P_k]$ that lie in at most one of $k$ residue classes, while a companion sequence $f_k$ counts the completely uncontained integers. It derives efficient, determinant-based recurrences $f_k=(p_k-1)f_{k-1}$ and $a_k=f_{k-1}+(p_k-1)a_{k-1}$, and proves that $a_k=av(k)$ and $f_k=free(k)$ by induction. The work also provides a novel determinant characterisation of the related sequence $A005867$ and shows how to generalise the framework to arbitrary sets of arithmetic progressions generated by $k$ distinct primes, with fast determinant computations. The results connect a classical number-theoretic counting problem to linear-algebraic structures and offer a scalable method for evaluating such determinants in practice.
Abstract
The sequence $A067549$ of The On-Line Encyclopedia of Integer Sequences is defined as $(a_k)_{k \geq 1}$ with $a_k$ being the determinant of the $k \times k$ matrix whose diagonal contains the first $k$ prime numbers and all other elements are ones. We relate this sequence to a concrete counting problem. Choose an arbitrary residue class $r_i$ for each prime $p_i$ with $1 \leq i \leq k$ and set $P_k = \prod_{i=1}^k p_i$. We show that $a_k$ is the number of integers in $[1, P_k]$ that are contained in \emph{at most} one of the $k$ chosen residue classes. Interestingly, we show that this sequence is closely related to the better known sequence $A005867$ for which we derive a novel characterisation in terms of determinants and which gives the number of integers in $[1, P_k]$ that are not contained in any of the $k$ residue classes. Our proof is purely structural and, therefore, it can be generalised to counting appearances of integers in residue classes of arbitrary arithmetic progressions generated by $k$ different primes using the determinant of a matrix of ones having those $k$ primes on its diagonal. The revealed structure also offers a fast way of calculating such determinants.