Problem 470

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?

Second 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 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?

Source: erdosproblems.com/470

Number theory
Divisors

Wiki page

Status

Open

No claim settles this problem.

Open. Under an unproved prime-gap hypothesis, Melfi proves that there are infinitely many primitive weird numbers (claim page (Melfi, 2014), conditional); the first question, on odd weird numbers, stays open.

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.

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.
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: Melfi, Giuseppe, On the conditional infiniteness of primitive weird numbers. J. Number Theory (2015), 508-514.

Problem 470

Second statement

Status

Notes

Proof claims

Discussion