Sum of squares of divisors of n

Conjecture: For each k = 2,3,..., all the rational numbers σ_k(n)/n^k = Σ_d|n 1/d^k (n = 1,2,3,...) have pairwise distinct fractional parts. - Zhi-Wei Sun, Oct 15 2015

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-oeis-1157,
  title        = {Sum of squares of divisors of n},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-1157}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Conjecture: For each k = 2,3,..., all the rational numbers $\frac{σ _{k} ( n )}{n ^{k}} = \sum_{d ∣ n} \frac{1}{d ^{k}}$ (n = 1,2,3,...) have pairwise distinct fractional parts. - Zhi-Wei Sun, Oct 15 2015

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«1157».

theorem conjecture :
  ∀ k : ℕ, 2 ≤ k →
    ∀ n₁ n₂ : ℕ, 0 < n₁ → 0 < n₂ → n₁ ≠ n₂ →
      Int.fract (↑((sigma k) n₁) / (↑n₁ ^ k : ℚ)) ≠
        Int.fract (↑((sigma k) n₂) / (↑n₂ ^ k : ℚ))

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

A001157

Source and licence

Imported from Formal Conjectures (OEIS), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.