Partial sums of powers of prime factors.

De Koninck, Jean-Marie; Luca, Florian

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Square-free Lucas $d$ -pseudoprimes and Carmichael-Lucas numbers

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Let $d$ be a fixed positive integer. A Lucas $d$ -pseudoprime is a Lucas pseudoprime $N$ for which there exists a Lucas sequence $U (P, Q)$ such that the rank of $N$ in $U (P, Q)$ is exactly $(N - ε (N)) / d$ , where $ε$ is the signature of $U (P, Q)$ . We prove here that all but a finite number of Lucas $d$ -pseudoprimes are square free. We also prove that all but a finite number of Lucas $d$ -pseudoprimes are Carmichael-Lucas numbers.