Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.6

Partial Sums of Powers of Prime Factors

Jean-Marie De Koninck
Département de Mathématiques et de Statistique
Université Laval
Québec G1K 7P4

Florian Luca
Mathematical Institute, UNAM
Ap. Postal 61-3 (Xangari)
CP 58 089
Morelia, Michoacán


Given integers $k\ge 2$ and $\ell\ge 3$, let $S_{k,\ell}^*$ stand for the set of those positive integers $n$ which can be written as $n=p_1^k+p_2^k+\ldots+p_\ell^k$, where $p_1,p_2,\ldots,p_\ell$ are distinct prime factors of $n$. We study the properties of the sets $S^*_{k,\ell}$ and we show in particular that, given any odd $\ell\ge 3$, $\displaystyle{\char93 \bigcup_{k=2}^\infty S_{k,\ell}^*=+\infty}$.

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Received January 5 2006; revised version received December 30 2006. Published in Journal of Integer Sequences December 30 2006.

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