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PRIME CHAINS AND PRATT TREES
"... ABSTRACT. We study the distribution of prime chains, which are sequences p1,..., pk of primes for which pj+1 ≡ 1 (mod pj) for each j. We first give conditional upper bounds on the length of Cunningham chains, chains with pj+1 = 2pj +1 for each j. We give estimates for P (x), the number of chains wit ..."
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ABSTRACT. We study the distribution of prime chains, which are sequences p1,..., pk of primes for which pj+1 ≡ 1 (mod pj) for each j. We first give conditional upper bounds on the length of Cunningham chains, chains with pj+1 = 2pj +1 for each j. We give estimates for P (x), the number of chains with pk � x (k variable), and P (x; p), the number of chains with p1 = p and pk � px. The majority of the paper concerns the distribution of H(p), the length of the longest chain with pk = p, which is also the height of the Pratt tree for p. We show H(p) � c log log p and H(p) � (log p) 1−c′ for almost all p, with c, c ′ explicit positive constants. We can take, for any ε> 0, c = e − ε assuming the ElliottHalberstam conjecture. A stochastic model of the Pratt tree is introduced and analyzed. The model suggests that for most p � x, H(p) stays very close to e log log x. 1.
ON COMMON VALUES OF φ(n) AND σ(m), II
"... Abstract. For each positiveinteger valued arithmetic function f, let Vf ⊂ N denote the image of f, and put Vf(x): = Vf ∩ [1,x] and Vf(x): = #Vf(x). Recently Ford, Luca, and Pomerance showed that Vφ ∩ Vσ is infinite, where φ denotes Euler’s totient function and σ is the usual sumofdivisors functio ..."
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Abstract. For each positiveinteger valued arithmetic function f, let Vf ⊂ N denote the image of f, and put Vf(x): = Vf ∩ [1,x] and Vf(x): = #Vf(x). Recently Ford, Luca, and Pomerance showed that Vφ ∩ Vσ is infinite, where φ denotes Euler’s totient function and σ is the usual sumofdivisors function. Work of Ford shows that Vφ(x) ≍ Vσ(x) as x → ∞. Here we prove a result complementary to that of Ford et al., by showing that most φvalues are not σvalues, and vice versa. More precisely, we prove that as x → ∞, #{n � x: n ∈ Vφ ∩ Vσ} � Vφ(x) + Vσ(x)
ON COMMON VALUES OF φ(n) AND σ(m), I
"... Abstract. We show, conditional on a uniform version of the prime ktuples conjecture, that there are x/(log x) 1+o(1) numbers not exceeding x common to the ranges of φ and σ. Here φ is Euler’s totient function and σ is the sumofdivisors function. 1. ..."
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Abstract. We show, conditional on a uniform version of the prime ktuples conjecture, that there are x/(log x) 1+o(1) numbers not exceeding x common to the ranges of φ and σ. Here φ is Euler’s totient function and σ is the sumofdivisors function. 1.
Article 13.8.8 On the Ratio of the Sum of Divisors and Euler’s Totient Function I
"... We prove that the only solutions to the equation σ(n) = 2 · φ(n) with at most three distinct prime factors are 3, 35 and 1045. Moreover, there exist at most a finite number of solutions to σ(n) = 2·φ(n) with Ω(n) ≤ k, and there are at most 22k +k −k squarefree solutions to φ(n) ∣ ∣σ(n) if ω(n) ..."
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We prove that the only solutions to the equation σ(n) = 2 · φ(n) with at most three distinct prime factors are 3, 35 and 1045. Moreover, there exist at most a finite number of solutions to σ(n) = 2·φ(n) with Ω(n) ≤ k, and there are at most 22k +k −k squarefree solutions to φ(n) ∣ ∣σ(n) if ω(n) = k. Lastly the number of solutions to φ(n) ∣ ∣σ(n) as x → ∞ is O ( xexp ( −1 √)) 2 logx. 1