Let $s'(n) = ∑_{d|n, 1First, we show that the count of quasiperfect n ≤ x is at most $x^{1/4+o(1)}$ as x → ∞. In fact, we show that for each fixed a, there are at most $x^{1/4+o(1)}$ natural numbers n ≤ x with σ(n) ≡ a (mod n) and σ(n) odd. (Quasiperfect n satisfy these conditions with a = 1.) For fixed δ ≠ 0, define the arithmetic function $s_{δ}(n) := σ(n) - n - δ$. Thus, s₁ = s'. Our second theorem says that the number of n ≤ x which are amicable with respect to $s_{δ}$ is at most $x/(log x)^{1/2+o(1)}$.
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Answering a question of Erdős, we show that a positive proportion of even numbers are in the form s(n), where s(n) = σ(n) - n, the sum of proper divisors of n.
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Let σ(n) denote the sum of positive divisors of the integer n, and let ϕ denote Euler's function, that is, ϕ(n) is the number of integers in the interval [1,n] that are relatively prime to n. It has been conjectured by Mąkowski and Schinzel that σ(ϕ(n))/n ≥ 1/2 for all n. We show that σ(ϕ(n))/n → ∞ on a set of numbers n of asymptotic density 1. In addition, we study the average order of σ(ϕ(n))/n as well as its range. We use similar methods to prove a conjecture of Erdős that ϕ(n-ϕ(n)) < ϕ(n) on a set of asymptotic density 1.
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Let λ denote Carmichael's function, so λ(n) is the universal exponent for the multiplicative group modulo n. It is closely related to Euler's φ-function, but we show here that the image of λ is much denser than the image of φ. In particular the number of λ-values to x exceeds $x/(log x)^{.36}$ for all large x, while for φ it is equal to $x/(log x)^{1+o(1)}$, an old result of Erdős. We also improve on an earlier result of the first author and Friedlander giving an upper bound for the distribution of λ-values.
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