Counting appearances of integers in sets of arithmetic progressions
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918254661861376 |
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| author | Pausinger, Florian |
| author_facet | Pausinger, Florian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_16358 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counting appearances of integers in sets of arithmetic progressions Pausinger, Florian Number Theory Combinatorics 11A07, 11B25 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. |
| title | Counting appearances of integers in sets of arithmetic progressions |
| topic | Number Theory Combinatorics 11A07, 11B25 |
| url | https://arxiv.org/abs/2512.16358 |