Counting appearances of integers in sets of arithmetic progressions

TL;DR

This paper explores a sequence related to prime-based determinants, linking it to counting integers in specific residue classes within arithmetic progressions, and introduces a structural, generalizable approach for such counting problems.

Contribution

It provides a novel determinant-based characterization of a sequence counting integers in at most one residue class, generalizes the counting to arbitrary progressions, and offers a fast determinant calculation method.

Findings

Sequence $A067549$ counts integers in at most one of k residue classes.

Established a link between $A067549$ and sequence $A005867$.

Developed a structural, generalizable method for counting in residue classes.

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.