A Remark on Relatively Prime Sets

Abstract

Four functions counting the number of subsets of {1, 2, . . . , n} having particular properties are defined by Nathanson and generalized by many authors. They derive explicit formulas for all four functions. In this paper, we point out that we need to compute only one of them as the others will follow as a consequence. Moreover, our method is simpler and leads to more general results than those in the literature.

Relatively Prime Sets, Divisor Sums, and Partial Sums

“…We see from (10) that the main terms can be obtained from the small value of d. Expanding the sum for d = 1, 2, 3, 4, we obtain…”

“…The functions f (n) and Φ (n) are introduced by Nathason [8] and generalized by many authors [2,3,9,11,12,15]. We refer the reader to Pongsriiam's article ( [9] or [10]) for a unified approach and the shortest calculation of the formulas for f (n), Φ (n) and their generalizations.…”