... contained in arithmetic progression r mod ρ, where (k − )r ≡ 0modρ.If d>ρthen every progression r mod dconsists of at most n/(ρ +1)elements hence it contains no more than 2n/(ρ+1)subsets. ... function x(t)=(k +1−(k −)t)/2. Note that x(1) = ( +1)/2,x(t2)=2andx(t1)=1, wheret2=k − 3k − ≥ 1andt1=k − 1k − .Furthermore, putH(x)=ln xx+x − 1xlnxx − 1.A note on ... Subject Classification: 11B75, 11P99.AsetA of positive integers is (k,)-sum-free for k, ∈ N, k>,ifthereare no solutions to the equation x1+ ···+ xk= y1+ ···+ yin A. Denoteby SFnk,the...