On common values of φ(n) and σ (n), II

Kevin Ford, Paul Pollack

Research output: Contribution to journalArticlepeer-review


For each positive-integer valued arithmetic function f, let V f ⊂ N denote the image of f, and put. Recently Ford, Luca, and Pomerance showed that Vφ ∩ Vσ is infinite, where ø denotes Euler's totient function and σ is the usual sum-of-divisors function. Work of Ford shows that Vø (x) {equivalent to} Vσ (x) as x → ∞. Here we prove a result complementary to that of Ford et al. by showing that most ø-values are not σ -values, and vice versa. More precisely, we prove that, as x → ∞.

Original languageEnglish (US)
Pages (from-to)1669-1696
Number of pages28
JournalAlgebra and Number Theory
Issue number8
StatePublished - 2012


  • Euler function
  • Sum of divisors
  • Totient

ASJC Scopus subject areas

  • Algebra and Number Theory


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