Abstract
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 language | English (US) |
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Pages (from-to) | 1669-1696 |
Number of pages | 28 |
Journal | Algebra and Number Theory |
Volume | 6 |
Issue number | 8 |
DOIs | |
State | Published - 2012 |
Keywords
- Euler function
- Sum of divisors
- Totient
ASJC Scopus subject areas
- Algebra and Number Theory