{
  "id": "2302.02243",
  "version": "v1",
  "published": "2023-02-04T21:09:01.000Z",
  "updated": "2023-02-04T21:09:01.000Z",
  "title": "Divisibility Properties of Integer Sequences",
  "authors": [
    "Daniel B. Shapiro"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "A sequence of nonzero integers $f = (f_1, f_2, \\dots)$ is ``binomid'' if every $f$-binomid coefficient $\\left[\\! \\begin{array}{c} n \\\\ k \\end{array}\\! \\right]_f$ is an integer. Those terms are the generalized binomial coefficients: \\[ \\left[\\! \\begin{array}{c} n \\\\ k \\end{array}\\! \\right]_f \\ = \\ \\frac{ f_nf_{n-1}\\cdots f_{n-k+1} }{ f_kf_{k-1}\\cdots f_1 }. \\] Let $\\Delta(f)$ be the infinite triangle with those numbers as entries. When $I = (1, 2, 3, \\dots)$ then $\\Delta(I)$ is Pascal's Triangle so that $I$ is binomid. Surprisingly, every row and column of Pascal's Triangle is also binomid. For any $f$, each row and column of $\\Delta(f)$ generates its own triangle and all those triangles fit together to form the ``Binomid Pyramid'' $\\mathbb{BP}(f)$. Sequence $f$ is ``binomid at every level'' if all entries of $\\mathbb{BP}(f)$ are integers. We prove that several familiar sequences have that property, including the Lucas sequences. In particular, $I = (1, 2, 3, \\dots )$, the sequence of Fibonacci numbers, and $(2^n - 1)_{n \\ge 1}$ are binomid at every level.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-04T21:09:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "integer sequences",
      "divisibility properties",
      "pascals triangle",
      "lucas sequences",
      "familiar sequences"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}