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 $$p_i^{a_i+1} < 2k(p_1p_2p_3\cdots p_k).$$ We conjecture that in fact one can always find an $i$ such that $p_i^{a_i+1} < 2p_1p_2p_3\cdots p_k$.