Partitions involving D.H. Lehmer numbers

Let n ≥ 2 be a fixed integer, let q and c be two integers with q > n and (n, q) = (c, q) = 1. For every positive integer a which is coprime with q we denote by `(a)]<sub>c</sub>{\overline{a}_{c}} the unique integer satisfying 1 ￡ `(a)]_c ￡ q{1\leq\overline{a}_{c} \leq{q}} and a`(a)]_c o c(mod q){a\overline{a}_{c} \equiv{c}({\rm mod}\, q)}. Put

L(q)={a ? Z+: (a,q)=1, n \not| a+`(a)]c }.L(q)=\{a\in{Z^{+}}: (a,q)=1, n {\not\hskip0.1mm|} a+\overline{a}_{c} \}.
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