Do any (2, 5)-perfect numbers exist

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $n\in\mathbb N$, write $\sigma(n)=\sum_{d\mid n,\ d>0}d$. The notation $(m,k)$-perfect means that $m$ successive applications of $\sigma$ multiply the original integer by $k$: $\sigma^{\circ m}(n)=kn$. In particular, $(2,5)$-perfect means $\sigma(\sigma(n))=5n$.

Statement recorded in the supplied source.

Does there exist $n\in\mathbb N$ such that \[\sigma(\sigma(n))=5n?\] [S2]

Short English statement

Does any positive integer have a sum of divisors whose own sum of divisors is five times the original integer?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400028 (AMR-093-0028). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Do any (2, 5)-perfect numbers exist. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.
• S3 Wikipedia contributors, Superperfect number, revision 1365374288 (2026) Generalizations: (m,k)-perfect numbers. https://en.wikipedia.org/w/index.php?title=Superperfect+number&oldid=1365374288.