Multiply perfect numbers in Lucas sequences with odd parameters (Q2714382)

Let \(P>0\) and \(Q\) be odd coprime integers such that \(P^2+4Q>0\), furthermore define sequences \(\{U_n\}\) and \(\{V_n\}\) by \(U_{n+2}=PU_{n+1}+QU_n\) and \(V_{n+2}=PV_{n+1}+QV_n\) \((n\geq 0)\) with initial terms \(U_0=0, U_1=1, V_0=2\) and \(V_1=P\). If \(\sigma (n)\) denotes the sum of the positive divisors of an integer \(n\) and \(\sigma (n)=kn\) for some integer \(k\), then \(n\) is called a multiply perfect number. In this paper the author proves that there exists an effectivelly computable constant \(C\) depending on \(P\) and \(Q\), such that if \(U_n\) or \(V_n\) is multiply perfect, then \(n<C\). Lemmas and propositions, which are used in the proof, are also interesting results.