Partial sums of (-1)^k - Ω(k)

"On May 16 2012, Zhi-Wei Sun conjectured that a(n) is positive for each n > 4." - _Zhi-Wei Sun_, May 19 2012

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.

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Cite

@misc{cairn-oeis-212496,
  title        = {Partial sums of (-1)^k - Ω(k)},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-212496}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-11}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

conjecture_1. "On May 16 2012, Zhi-Wei Sun conjectured that $a (n)$ is positive for each $n > 4$ ."

_Zhi-Wei Sun_, May 19 2012

conjecture_2. "Moreover, he guessed that $a (n) > n$ for any $n > 324$ ."

_Zhi-Wei Sun_, May 19 2012

conjecture_3. "He also guessed that $a (n) < n lo g (lo g (n))$ for $n > 5892$ ."

_Zhi-Wei Sun_, May 19 2012

conjecture_4. "Sun also conjectured that $b (n) = \sum_{k = 1}^{n} (- 1)^{k - Ω (k)} / k < 0$ for all $n = 1, 2, 3, \dots$ "

_Zhi-Wei Sun_, May 19 2012

conjecture_5. "Moreover, he guessed that $b (n) < - 1/ n$ for all $n > 1$ ."

_Zhi-Wei Sun_, May 19 2012

conjecture_6. "He also guessed that $b (n) > - lo g (lo g (n)) / n$ for $n > 2008$ ."

_Zhi-Wei Sun_, May 19 2012

The sequence $a (n) = \sum_{k = 1}^{n} (- 1)^{k - Ω (k)}$ , where $Ω (k)$ is the total number of prime factors of $k$ counted with multiplicity.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«212496» (6 statements).

theorem conjecture_1 (n : ℕ) (hn : 4 < n) : 0 < a n

theorem conjecture_2 (n : ℕ) (hn : 324 < n) :
    Real.sqrt (n : ℝ) < (a n : ℝ)

theorem conjecture_3 (n : ℕ) (hn : 5892 < n) :
    (a n : ℝ) < Real.sqrt (n : ℝ) * Real.log (Real.log (n : ℝ))

theorem conjecture_4 (n : ℕ) (hn : 0 < n) : b n < 0

theorem conjecture_5 (n : ℕ) (hn : 1 < n) :
    (b n : ℝ) < -1 / Real.sqrt (n : ℝ)

theorem conjecture_6 (n : ℕ) (hn : 2008 < n) :
    -Real.log (Real.log (n : ℝ)) / Real.sqrt (n : ℝ) < (b n : ℝ)

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
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

A212496

Source and licence

Imported from Formal Conjectures (OEIS), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.