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Licensed Unlicensed Requires Authentication Published by De Gruyter November 5, 2010

On Relatively Prime Subsets and Supersets

Mohamed El Bachraoui
From the journal

Abstract

A nonempty finite set of positive integers A is relatively prime if gcd(A) = 1 and it is relatively prime to n if gcd(A ∪ {n}) = 1. The number of nonempty subsets of A which are relatively prime to n is Φ(A, n) and the number of such subsets of cardinality k is Φk(A, n). Given positive integers l1, l2, m2, and n such that l1l2m2 we give Φ([1, m1] ∪ [l2, m2], n) along with Φk([1, m1] ∪ [l2, m2], n). Given positive integers l, m, and n such that lm we count for any subset A of {l, l + 1, . . . , m} the number of its supersets in [l, m] which are relatively prime and we count the number of such supersets which are relatively prime to n. Formulas are also obtained for corresponding supersets having fixed cardinalities. Intermediate consequences include a formula for the number of relatively prime sets with a nonempty intersection with some fixed set of positive integers.

Received: 2009-10-09
Accepted: 2010-05-17
Published Online: 2010-11-05
Published in Print: 2010-November

© de Gruyter 2010