{
  "id": "2311.14021",
  "version": "v1",
  "published": "2023-11-23T14:21:39.000Z",
  "updated": "2023-11-23T14:21:39.000Z",
  "title": "The fourth positive element in the greedy $B_h$-set",
  "authors": [
    "Melvyn B. Nathanson",
    "Kevin O'Bryant"
  ],
  "comment": "7 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "For $h \\geq 1$, a $B_h$-set is a set of integers such that every integer $n$ has at most one representation in the form $n = a_{i_1} + \\cdots + a_{i_h}$, where $a_{i_r} \\in A$ for all $r = 1,\\ldots, h$ and $a_{i_1} \\leq \\ldots \\leq a_{i_h}$. The greedy $B_h$-set is the infinite set of nonnegative integers $\\{a_0(h), a_1(h), a_2(h), \\ldots \\}$ constructed as follows: If $a_0(h) = 0$ and $\\{a_0(h), a_1(h), a_2(h), \\ldots, a_k(h) \\}$ is a $B_h$-set, then $a_{k+1}(h)$ is the least positive integer such that $\\{a_0(h), a_1(h), a_2(h), \\ldots, a_k(h), a_{k+1}(h) \\}$ is a $B_h$-set. Then $a_1(h) = 1$, $a_2(h) = h+1$, and $a_3(h) = h^2+h+1$ for all $h$. This paper proves that $a_4(h)$, the fourth term of the greedy $B_h$-set is $\\left( h^3 + 3h^2 + 3h + 1\\right) /2$ if $h$ is odd and $\\left( h^3 + 2h^2 + 3h + 2\\right) /2$ if $h$ is even.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-23T14:21:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B13",
      "11B34",
      "11B75",
      "11P99"
    ],
    "keywords": [
      "fourth positive element",
      "infinite set",
      "fourth term",
      "nonnegative integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}