Must a primitive nondeficient number have a component not much larger than its radical?
Abstract
Let $n$ be a primitive nondeficient number where $n=p_1^{a_1}p_2^{a_2} \cdots p_k^{a_k}$ where $p_1, p_2 \cdots p_k$ are distinct primes. We prove that there exists an $i$ such that $$p_i^{a_i+1} < 2k(p_1p_2p_3\cdots p_k).$$ We conjecture that in fact one can always find an $i$ such that ${p_i}^{a_i+1} < p_1p_2p_3\cdots p_k$.
 Publication:

arXiv eprints
 Pub Date:
 May 2020
 DOI:
 10.48550/arXiv.2005.12115
 arXiv:
 arXiv:2005.12115
 Bibcode:
 2020arXiv200512115Z
 Keywords:

 Mathematics  Number Theory;
 11A25;
 11N64
 EPrint:
 8 pages. This new version has substantially tighter bounds due to a suggestion by JanChristoph SchlagePuchta