Odd Weird Numbers

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

A positive integer $n$ is abundant when $\sigma(n)>2n$. It is semiperfect when some subset of its proper positive divisors sums to $n$. A weird number is abundant but not semiperfect. The target asks for an odd integer with both properties; the full divisor sum alone is not enough. [S1] [S2]

Statement recorded in the supplied source.

Do any odd weird numbers exist? [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Construct an odd weird number or rule out all odd integers. [S1]

Short English statement

Can an odd integer have more than enough proper-divisor mass to reach itself, while no subset of those divisors sums to it exactly?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 1257 (NT-050), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 J. Fang, Searching on the boundary of abundance for odd weird numbers, arXiv:2207.12906. https://arxiv.org/abs/2207.12906. Bibliographic information from the supplied record.

• S3 T. F. Bloom, Erdős Problem #470 (updated 2026). https://www.erdosproblems.com/470. Bibliographic information from the supplied record.