Let $s(n): = ∑_{d|n, d Let ℓ ≥ 1. A natural number n is said to be ℓ-full (or ℓ-powerful) if $p^{ℓ}$ divides n whenever the prime p divides n. As shown by Erdős and Szekeres in 1935, the number of ℓ-full n ≤ x is asymptotically $c_{ℓ} x^{1/ℓ}$, as x → ∞. Here $c_{ℓ}$ is a positive constant depending on ℓ. We show that for each fixed ℓ, the set of amicable ℓ-full numbers has relative density zero within the set of ℓ-full numbers.
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In 1909, Hilbert proved that for each fixed k, there is a number g with the following property: Every integer N ≥ 0 has a representation in the form N = x 1k + x 2k + … + x gk, where the x i are nonnegative integers. This resolved a conjecture of Edward Waring from 1770. Hilbert’s proof is somewhat unsatisfying, in that no method is given for finding a value of g corresponding to a given k. In his doctoral thesis, Rieger showed that by a suitable modification of Hilbert’s proof, one can give explicit bounds on the least permissible value of g. We show how to modify Rieger’s argument, using ideas of F. Dress, to obtain a better explicit bound. While far stronger bounds are available from the powerful Hardy-Littlewood circle method, it seems of some methodological interest to examine how far elementary techniques of this nature can be pushed.
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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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