  We address the primary decomposition of the knot concordance group in terms
of the solvable filtration and higher-order von Neumann $\rho$-invariants by
Cochran, Orr, and Teichner. We show that for a nonnegative integer n, if the
connected sum of two n-solvable knots with coprime Alexander polynomials is
slice, then each of the knots has vanishing von Neumann $\rho$-invariants of
order n. This gives positive evidence for the conjecture that nonslice knots
with coprime Alexander polynomials are not concordant. As an application, we
show that if K is one of Cochran-Orr-Teichner's knots which are the first
examples of nonslice knots with vanishing Casson-Gordon invariants, then K is
not concordant to any knot with Alexander polynomial coprime to that of K.
