{
  "abstract": "Ranjan and Singh proved that for every composite $n$ which is not twice an odd prime, $φ(n)$ is at least one plus the sum of $φ$ over the prime divisors of $n$, with equality exactly for $n=4,12,30$. \\new{We study the weighted equation $φ(n)=c+α\\sum_{p\\mid n}φ(p)$. We determine the parameters $(c,α)$ admitting infinite families of solutions, prove that all remaining solutions are finite in number and arise from an explicit linear mechanism, and compute them exactly in a range of parameters; the finiteness of $\\{4,12,30\\}$ becomes a special case. We also settle the analogous equations for the divisor functions $τ$ and $σ$ and for the Jordan totients, and, from known results on the sum of the prime divisors, derive the full asymptotic expansion of the average deficiency.} \\K{In the range $|c|\\le10$, $1\\leα\\le6$ the composite solution set is determined completely\\V{, and the number and the size of the sporadic solutions are shown to be bounded by polynomials in $|c|+α$}.} \\B{Finally, replacing the prime values $φ(p)$ by their iterates $φ^{(k)}(p)$, we determine all solutions of $φ(n)=1+\\sum_{p\\mid n}φ^{(k)}(p)$: the solution sets are $\\{1,4,12,30\\}$, $\\{1,3,4,10\\}$ and, for every $k\\ge3$, $\\{1,3,4\\}$; for the weighted iterated equation we exhibit an infinite family governed by the safe primes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36260",
  "author_total": 3,
  "authors": [
    {
      "name": "Hacène Belbachir"
    },
    {
      "name": "Ravi Ranjan"
    },
    {
      "name": "Shubh N. Singh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/99zwn31hk4fe5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36260",
  "fulltext_url": null,
  "id": "99zwn31hk4fe5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36260",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7224098924",
  "pdf_url": "https://arxiv.org/pdf/2610.36260",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/99zwn31hk4fe5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36260",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36260"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Infinite families and sporadic solutions of the equation $φ(n)=c+α\\sum_{p\\mid n}φ(p)$",
  "title_text": "Infinite families and sporadic solutions of the equation φ(n)=c+α∑_p| nφ(p)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/99zwn31hk4fe5",
  "venue": "arXiv",
  "versions": []
}