Consecutive coincidences of Euler's function (Q2805968)

Let \({\mathcal A} = \{ n \in {\mathbb N} : \varphi (n) = \varphi (n+1)\}\) and \({\mathcal A}(x) = |{\mathcal A} \bigcap [1,x] |\). The paper under review deals with the sum \(S:= \sum \limits_{n \in {\mathcal A}} \frac {1}{n}\). More precisely, the authors establish that NEWLINE\[CARRIAGE_RETURNNEWLINE 1.4324884 < S < 441702. CARRIAGE_RETURNNEWLINE\]NEWLINE The main tools in proving this result are a version of the Hardy-Ramanujan inequality, obtaining an economical and explicit upper bound for \(\pi_k(x)\), the number of \(n \leq x\) with exactly \(k\) distinct prime factors when \(x \geq 10^{12}\), \(k \geq 1\), and an explicit bound for the counting function of \(y\)-smooth numbers upto \(x\) for \(x \geq 2973\), \(y \geq 200\). The results are quite interesting and even have independent interest.NEWLINENEWLINEFor related references, the readers are referred to [\textit{J. Bayless} and \textit{D. Klyve}, Integers 11, No. 3, 315--332, A5 (2011; Zbl 1259.11010); \textit{P. Erdős}, Proc. Camb. Philos. Soc. 32, 530--540 (1936; JFM 62.1149.02); Publ. Math. 4, 108--112 (1955; Zbl 0065.02706); \textit{P. Erdős} et al., Acta Math. Hung. 49, 251--259 (1987; Zbl 0609.10034); \textit{S. W. Graham} et al., in: Number theory in progress. Volume 2: Elementary and analytic number theory. Berlin: de Gruyter. 867--882 (1999; Zbl 0937.11037); \textit{H. Nguyen}, The reciprocal sum of the amicable numbers. Dartmouth College (2014); with \textit{C. Pomerance}, Math. Comput. 88, No. 317, 1503--1526 (2019; Zbl 1429.11009); \textit{C. Pomerance}, J. Reine Angew. Math. 325, 183--188 (1981; Zbl 0448.10007); \textit{C. Pomerance} (ed.) and \textit{M. Th. Rassias} (ed.), Analytic number theory. In honor of Helmut Maier's 60th birthday. Cham: Springer (2015; Zbl 1336.11002)] and [\textit{T. Yamada}, ``On equations \(\sigma(n)=\sigma(n+k)\) and \(\varphi(n)=\varphi(n+k)\)'', Preprint, \url{arXiv:1001.251121}, J. Comb. Number Theory 9, No. 1, 15--21 (2017; Zbl 1429.11013)].