Problem 470

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claims/: The 1 claim page of Problem 470, one per claimant's result; the problem's standing derives from them.

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Statement. Call $n$ weird if $\sigma(n)\geq 2n$ and $n$ is not pseudoperfect, that is, it is not the sum of any set of its divisors.

Are there any odd weird numbers? Are there infinitely many primitive weird numbers, i.e. those such that no proper divisor of $n$ is weird?

Statement (corrected). Call $n$ weird if $\sigma(n)\geq 2n$ and $n$ is not pseudoperfect, that is, it is not the sum of any set of distinct proper divisors of $n$.

Are there any odd weird numbers? Are there infinitely many primitive weird numbers, i.e. those such that no proper divisor of $n$ is weird?

Notes. Read as the site words it, the gloss of pseudoperfect allows the one-element set $\{n\}$. That would make every $n$ pseudoperfect and both questions trivially negative. The corrected Statement replaces "any set of its divisors" by "any set of distinct proper divisors of $n$"; nothing else changes. The sources read it with proper divisors. Erdős and Graham (1980), p. 94, the site's source for the wording, call $n$ weird when $\sigma(n)/n\ge2$ and $n$ is not a sum of distinct proper divisors of $n$. Benkoski and Erdős (1974), p. 617 (card), define pseudoperfect that way and abundant as $\sigma(n)\ge2n$. The formal-conjectures statement also uses proper divisors.

Formulation. The formal-conjectures statement uses strict abundance. Since perfect numbers are pseudoperfect, $\ge$ and $>$ give the same weird numbers.

Status. Open. Under an unproved prime-gap hypothesis, Melfi proves that there are infinitely many primitive weird numbers (claim page, conditional); the first question, on odd weird numbers, stays open.

Source. erdosproblems.com/470, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #470, https://www.erdosproblems.com/470.

References.

• [BeEr74] Benkoski, S. J. and Erdős, P., On weird and pseudoperfect numbers. Math. Comp. (1974), 617-623.
• [Fa22] Searching on the boundary of abundance for odd weird numbers, W. Fang. arXiv:2207.12906 (2022).
• [Gu04] Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp.; doi:10.1007/978-0-387-26677-0. Section B2 "Almost perfect, quasi-perfect, pseudoperfect, harmonic, weird, multiperfect and hyperperfect numbers", printed p. 77, where the book defines weird numbers and poses the odd and primitive weird questions. Library home: guy_2004_unsolved_problems_number_theory.
• [LiRi18] J. Liddy and J. Riedl, An algorithm to determine all odd primitive abundant numbers with $d$ prime divisors. Honors Research Projects. 728 (2018).
• [Me15] [[../library/divisors/melfi_2015_conditional_infiniteness_primitive_weird_numbers/_index|Melfi, Giuseppe, On the conditional infiniteness of primitive weird numbers]]. J. Number Theory (2015), 508-514.

Formalization. Statement in formal-conjectures.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• fang_2022_searching_boundary_abundance_odd_weird_numbers
• fang_2022_searching_boundary_abundance_odd_weird_numbers / theorem_1_1
• fang_2022_searching_boundary_abundance_odd_weird_numbers / theorem_1_2
• melfi_2015_conditional_infiniteness_primitive_weird_numbers
• melfi_2015_conditional_infiniteness_primitive_weird_numbers / theorem_1
• guy_2004_unsolved_problems_number_theory