E095: Squarefree filter: ω(n)=Ω(n) when μ(n)≠0

E095 hero

E095: Squarefree filter: ω(n)=Ω(n) when μ(n)≠0

Tags: number-theory, model-checking, visualization, mobius, omega, arithmetic-functions

Highlights

  • Focused numeric experiment with a small set of figures.

  • Parameters saved to params.json for reproducibility.

  • Defaults are chosen, so the experiment remains feasible for the CI “slow” suite.

What is computed

  • Use μ(n) as a squarefree indicator and verify ω(n)=Ω(n) on that subset.

Notes

  • This page summarizes the intent; see the generated report.md in out/ for concrete outputs.

Published run snapshot

If this experiment is included in the docs gallery, include the published snapshot (report + params).

Artifacts

  • figures/fig_01_bigomega_minus_omega.png

  • params.json

  • report.md

Notes

  • Fraction with Ω(n)=ω(n) on all integers: 0.6079

  • Fraction with Ω(n)=ω(n) on squarefree integers: 1.0000 (should be 1.0)

  • Ω(n)-ω(n) counts repeated prime powers (p^k with k≥2).

params.json (snapshot)
{
  "max_k": 12,
  "n_max": 200000
}

References

  • See the arithmetic-functions background pages in docs/background/.