Odd weird numbers

Find an odd number whose proper divisors sum to more than it but no subset sums to it exactly, or prove none exists.

shoal State the problem, from Erdős Problems 039aae2 · 2周前历史

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1	# Odd weird numbers
2
3	A number is weird if its proper divisors add up to more than the number itself, yet no selection of them adds up to exactly the number. The smallest weird number is $70$, and weird numbers have positive density, but every known example is even. Benkoski and Erdős asked whether an odd weird number exists.
4
5	## Problem
6
7	Let $\sigma(n)$ be the sum of all divisors of $n$. Call $n$ pseudoperfect (or semiperfect) if $n$ is the sum of some set of distinct proper divisors of $n$, and weird if $\sigma(n)\ge 2n$ but $n$ is not pseudoperfect.
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9	1. Is there an odd weird number?
10	2. Are there infinitely many primitive weird numbers, that is, weird numbers none of whose proper divisors is weird?
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12	Erdős offered \$10 for settling the first question.
13
14	## Background
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16	Weird numbers were studied by Benkoski and Erdős in 1974, who proved that they have positive density. For example, $70$ is weird: its proper divisors $1,2,5,7,10,14,35$ sum to $74$, but no subset of them sums to $70$. Odd abundant numbers exist (the smallest is $945$), but every odd abundant number below $10^{21}$ is pseudoperfect. The weird numbers form sequence A006037 in the OEIS, and the problem is B2 in Guy's Unsolved Problems in Number Theory.
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18	## What is known
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20	- Searches. Fang (2022) showed that there is no odd weird number below $10^{21}$, and none below $10^{28}$ among numbers whose abundance $\sigma(n)-2n$ is below $10^{14}$.
21	- Structure. Liddy and Riedl (2018) showed that an odd weird number must have at least six distinct prime factors.
22	- Primitive weird numbers. Melfi (2015) proved that there are infinitely many primitive weird numbers, assuming that consecutive primes satisfy $p_{n+1}-p_n<\tfrac1{10}p_n^{1/2}$ for all large $n$; this hypothesis would follow from standard conjectures on prime gaps.
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24	## What would count as progress
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26	- An odd weird number, given with its factorisation; that it is abundant and not pseudoperfect can be checked by a subset-sum computation over its divisors.
27	- An extension of the search beyond $10^{21}$, with code and a description of the pruning so that the claimed range can be audited.
28	- An unconditional proof that there are infinitely many primitive weird numbers, or new necessary conditions on an odd weird number.
29	- A formal proof against the Lean statement in [formal-conjectures](https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/470.lean).
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31	## References
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33	1. S. J. Benkoski and P. Erdős, On weird and pseudoperfect numbers, Math. Comp. 28 (1974), 617–623. [doi:10.1090/S0025-5718-1974-0347726-9](https://doi.org/10.1090/S0025-5718-1974-0347726-9)
34	2. G. Melfi, On the conditional infiniteness of primitive weird numbers, J. Number Theory 147 (2015), 508–514. [doi:10.1016/j.jnt.2014.07.024](https://doi.org/10.1016/j.jnt.2014.07.024)
35	3. W. Fang, Searching on the boundary of abundance for odd weird numbers (2022). [arXiv:2207.12906](https://arxiv.org/abs/2207.12906)
36	4. OEIS Foundation, Weird numbers, sequence A006037. [oeis.org/A006037](https://oeis.org/A006037)
37
38	## Source
39
40	Imported from [Erdős Problems 470](https://www.erdosproblems.com/470), curated by Thomas Bloom. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.