Journal of Integer Sequences, Vol. 29 (2026), Article 26.4.3

Must a Primitive Non-Deficient Number Always Have a Small Component?


Joshua Zelinsky
Department of Mathematics
Hopkins School
New Haven, CT 06515
USA

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 $p_1, p_2, \ldots, p_k$ are distinct primes. We prove that there exists an $i$ such that

$\displaystyle p_i^{a_i+1} < 2k(p_1p_2p_3\cdots p_k).$

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

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(Concerned with sequences A000396 A005100 A006039.)

Received December 10 2024; revised versions received March 13 2026; March 14 2026; June 19 2026. Published in Journal of Integer Sequences, July 3 2026.

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