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The number of relatively prime subsets of a finite union of sets of consecutive integers. (English) Zbl 1285.11006

Summary: Let \(A\) be a finite union of disjoint sets of consecutive integers and let \(n\) be a positive integer. We give a formula for the number of relatively prime subsets (resp., relatively prime subsets of cardinality \(k\)) of \(A\), which generalizes results of M. B. Nathanson and B. Orosz [Integers 7, No. 1, A54, 5 p. (2007; Zbl 1159.11010)], M. El Bachraoui [Integers 7, No. 1, A43, 8 p. (2007; Zbl 1139.11005)] and others. We give as well similar formulas for the number of subsets with gcd coprime to \(n\).

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
11B05 Density, gaps, topology
11B75 Other combinatorial number theory
11D41 Higher degree equations; Fermat’s equation
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Online Encyclopedia of Integer Sequences:

Number of subsets of {1,2,...,n} with relatively prime elements.