Loeschian Numbers Revisited

 I've blogged about Loeschian numbers in the following posts:

• Loeschian Numbers on the 5th January 2022
• Loeschian Primes on the 21st May 2023
• Linear Relationships Between Loeschian Numbers on the 11th June 2023

Here is some additional information provided by Gemini:

Approximately 19.63% of the positive integers between 1 and 40000 are Loeschian numbers. This equates to roughly 7851 integers in that range.

Why the Percentage Drops Over Time? You might intuitively expect numbers representable by $i^2 + i \, j + j^2$ to maintain a steady density, but they actually become progressively sparser as you move higher up the number line.

In 1975, economic geographer J. U. Marshall tabulated that between 1 and 10000 inclusive, there are exactly 2299 Loeschian numbers—a density of 22.99% [7.1.5]. By the time you reach 40000, that density drops by over three percentage points.

This thinning behavior is governed by a Landau-Ramanujan-like asymptotic law. The total count of Loeschian numbers $N(x)$ up to a threshold $x$ grows according to the formula:$$N(x) \sim \alpha \frac{x}{\sqrt{\ln x}}$$

Here, $\alpha$ is the Loeschian density constant (OEIS A301429), which is approximately $0.6389094$. Because the denominator $\sqrt{\ln x}$ grows continuously as $x$ increases, the overall percentage of Loeschian numbers steadily decays toward zero as you approach infinity.

The total Loeschian numbers up to 1000000 is 180874, a percentage 18.09% (permalink).

I've now incorporated whether a number is Loeschian or not into my daily number analysis (permalink) as shown below where \(28225=15^2+15 \times 160 + 160^2\).