Compositions with Pairwise Relatively Prime Summands within a Restricted Setting

The paper studies the counting function R2(n, k) = ∑ a1+a2+···+ak=n (ai,aj)=1 i 6=j 1, ai ≥ 1, k ≥ 2 with ai, n and k positive integers and establishes a relationship between R2(n, k) and P2(n, k) where P2(n, k) = ∑ a1+a2+···+ak=n 1≤a1≤a2≤···≤ak≤n (ai,aj)=1 i 6=j 1, ai ≥ 1, k ≥ 2 with ai, n, k positive integers. 1. PRELIMINARIES Gould [4], studied the number-theoretic function R(n, k) = ∑ a1+a2+···+ak=n (a1,a2,··· ,ak)=1 1, ai ≥ 1, k ≥ 2 (1) and showed amongst other results that: Theorem 1: R(n, k) = ∑ d|n ( d− 1 k − 1 ) μ( n d ) and Theorem 2: ∞ ∑ j=1 R(j, k) x 1− xj = x (1− x)k . Motivated by definition 1, we now define R2(n, k) = ∑ a1+a2+···+ak=n (ai,aj)=1 i 6=j 1, ai ≥ 1, k ≥ 2. (2)