Even perfect numbers among generalized Fibonacci sequences

Abstract

A positive integer \(n\) is said to be a perfect number, if \(\sigma (n)=2n\), where \(\sigma (N)\) is the sum of all positive divisors of \(N\). Luca (Rend Circ Mat Palermo Ser II 49:313–318, 2000) proved that there is no perfect number in the Fibonacci sequence. For \(k\ge 2\), the \(k\)-generalized Fibonacci sequence \((F_n^{(k)})_{n}\) is defined by the initial values \(0,0,\ldots ,0,1\) (\(k\) terms) and such that each term afterwards is the sum of the \(k\) preceding terms. In this paper, we prove, among other things, that there is no even perfect numbers belonging to \(k\)-generalized Fibonacci sequences when \(k\not \equiv 3\pmod 4\).