Asymptotic Density of Powerful Numbers

[franzhusch]

What is the conjecture

A positive integer $n$ is powerful if for every prime $p$ dividing $n$, we have $p^2 \mid n$. Let $Q(x)$ denote the number of powerful integers up to $x$. The Asymptotic Density of Powerful Numbers conjecture states that:

$$Q(x) = \zeta(1/2)^{-1} \sqrt{x} + o(\sqrt{x})$$

where $\zeta$ is the Riemann zeta function and $\zeta(1/2)^{-1} \approx 0.3039$. Equivalently, the natural density of powerful numbers is $\zeta(1/2)^{-1}$.

(This description may contain subtle errors especially on more complex problems; for exact details, refer to the sources.)

Sources:

• https://en.wikipedia.org/wiki/Powerful_number, https://mathworld.wolfram.com/PowerfulNumber.html, Bateman, P. T., & Knopp, M. I. (1975). Some problems concerning the prime factors of consecutive integers. Acta Arith., https://oeis.org/A001694

Prerequisites needed

Formalizability Rating: 2/5 (0 is best) (as of 2026-04-03)

Building blocks (from search results):

• Nat.Prime and divisibility theory (foundational in Mathlib)
• Complex.zeta function (available in Mathlib)
• Asymptotic notation o(...) (available via Asymptotics in Mathlib)

Missing pieces:

• IsPowerful predicate definition for integers where all prime factors appear with exponent ≥ 2 (not in Mathlib; similar to Powerfree in formal-conjectures repo but inverse property)
• Formal asymptotic density definition and convergence theory for counting functions (requires new definitions connecting divisor/factor theory to asymptotic analysis)

Rating justification: The core definitions (primes, divisibility, zeta function) are present in Mathlib, but formalizing powerful numbers as a predicate and connecting the counting function $Q(x)$ to asymptotic analysis requires moderate infrastructure development beyond simple composition of existing concepts.

AMS categories

• ams-11

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• I plan on adding this conjecture to the repository
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