Note on some congruences of Lehmer

AbstractIn the paper, we generalize some congruences of Lehmer and prove that for any positive integer n with (n,6)=1∑r=1(r,n)=1⌊n/3⌋1n−3r≡12qn(3)−14nqn2(3)(modn2),∑r=1(r,n)=1⌊n/4⌋1n−4r≡34qn(2)−38nqn2(2)(modn2) and∑r=1(r,n)=1⌊n/6⌋1n−6r≡13qn(2)+14qn(3)∑r=1(r,n)=1⌊n/6⌋1n−6r≡−n(16qn2(2)+18qn2(3))(modn2), where qn(a)=(aϕ(n)−1)/n