Must a primitive non-deficient number have a component not much larger than its radical?

Abstract

Let $n$ be a primitive non-deficient 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 $$p_i^{a_i+1}<2k(p_1{p}_2{p}_3\cdots p_k).$$ We conjecture that in fact one can always find an $i$ such that $p_i^{a_i+1}<2p_1{p}_2{p}_3\cdots p_k$.

arXiv comment14 pages. This version tighter bounds than the first version due to a suggestion by Jan-Christoph Schlage-Puchta. A hole in one proof was cleaned up, and Conjecture 1 has been been weakened. Now appearing in the Journal of Integer Sequences

References in corpus (1)

• The reciprocal sum of primitive nondeficient numbers