On Relatively Prime Subsets and Supersets
نویسنده
چکیده
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 l1 ≤ l2 ≤ m2 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 l ≤ m 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.
منابع مشابه
THE NUMBER OF RELATIVELY PRIME SUBSETS AND PHI FUNCTIONS FOR { m , m + 1 , . . . , n } Mohamed
The work in this paper is inspired and motivated by some work of Nathanson. We count the number of relatively prime subsets and the number of relatively prime subsets having some fixed cardinality that are in {m,m+1, . . . , n}. We also count the number of nonempty subsets of {m,m+1, . . . , n} whose gcd is relatively prime to n and the number of nonempty subsets {m,m+1, . . . , n} having some ...
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