Note: Posting in MO since it was unanswered in MSE

Definition: We say that a natural number $n$ has triangular divisors if it has at least one triplet of divisors $n = d_1d_2d_3, 1 \le d_1 \le d_2 \le d_3$, such that $d_1,d_2$ and $d_3$ form the sides of a non-degenerate triangle.

Eg: $60$ has triangular divisors because $60 = 3.4.5$ and $3,4,5$ form a triangle. Note that another triplet of divisors of $60 = 1.4.15$ does not form a triangle but because of the triplet $3,4,5$ the number $60$ qualifies a number with triangular divisors. On the other the number $10$ does not have any triplet of triangular divisors.

The first few numbers in this sequence are

$$ 1,4,8,9,12,16,18,24,25,27,32,36,40,45,48,\ldots $$

I am interested in the density of these numbers. In the linked questions users commented that the almost all integers are expected this property since because most numbers will have a several small prime factors so the natural density was initially thought to be $1$.

However, quite counter intuitively, in one of the long comment which was posted as an answer in the linked question, it was proved that if $n$ has triangular divisors then the largest prime factor of $n$ is less than $\sqrt n$ which immediately implies that the natural density of numbers with triangular divisors is $ < 1 - \log 2 \approx 0.3069$. Experimentally, the data shows that the natural density approaches $0$. Let $f(x)$ be the number of integers $\le x$ with triangular divisors. We have $$ f(46732002) = 3630678 $$ The graph of $\frac{f(x)}{x}$ vs $x$ given below shows that the density decreases as $x$ increases.

Experimental data: Let $f(x)$ be the number of integers $\le x$ with this property. The graph of $\frac{f(x)}{x}$ vs. $x$ is shown below.

A simple curve fitting gives $\frac{a}{\log x}$ as a good fit with $R^2 = 0.9977$ which suggests that $f(x)$ growth rate somewhere close to $\pi(x)$. This is rather counter intuitive as mentioned in the comments that we expect almost all integers to have this property.

Higher density of even numbers: A curious observation is the there are significantly more even numbers with triangular divisors as compared to odd numbers. Let $f_o(x)$ be the number of odd numbers $\le x$ with triangular divisors. The graph of $\frac{f_o(x)}{f(x)}$ is shown below.

Question: How many numbers $\le x$ have triangular divisors and why are even numbers more dense than odd numbers?

Related question: Reshaping an object into two integer sided cuboids without changing the total volume.

Note: Posting in MO since it was unanswered in MSE

Definition: We say that a natural number $n$ has triangular divisors if it has at least one triplet of divisors $n = d_1d_2d_3, 1 \le d_1 \le d_2 \le d_3$, such that $d_1,d_2$ and $d_3$ form the sides of a non-degenerate triangle.

Eg: $60$ has triangular divisors because $60 = 3.4.5$ and $3,4,5$ form a triangle. Note that another triplet of divisors of $60 = 1.4.15$ does not form a triangle but because of the triplet $3,4,5$ the number $60$ qualifies a number with triangular divisors. On the other the number $10$ does not have any triplet of triangular divisors.

The first few numbers in this sequence are

$$ 1,4,8,9,12,16,18,24,25,27,32,36,40,45,48,\ldots $$

I am interested in the density of these numbers. In the linked questions users commented that the almost all integers are expected this property since because most numbers will have a several small prime factors so the natural density was initially thought to be $1$.

However, quite counter intuitively, in one of the long comment which was posted as an answer in the linked question, it was proved that if $n$ has triangular divisors then the largest prime factor of $n$ is less than $\sqrt n$ which immediately implies that the natural density of numbers with triangular divisors is $ < 1 - \log 2 \approx 0.3069$. Experimentally, the data shows that the natural density approaches $0$. Let $f(x)$ be the number of integers $\le x$ with triangular divisors. We have $$ f(46732002) = 3630678 $$ The graph of $\frac{f(x)}{x}$ vs $x$ given below shows that the density decreases as $x$ increases.

Experimental data: Let $f(x)$ be the number of integers $\le x$ with this property. The graph of $\frac{f(x)}{x}$ vs. $x$ is shown below.

A simple curve fitting gives $\frac{a}{\log x}$ as a good fit with $R^2 = 0.9977$ which suggests that $f(x)$ growth rate somewhere close to a constant times the prime counting function $\pi(x)$. This is rather counter intuitive as mentioned in the comments that we expect almost all integers to have this property.

Higher density of even numbers: A curious observation is the there are significantly more even numbers with triangular divisors as compared to odd numbers. Let $f_o(x)$ be the number of odd numbers $\le x$ with triangular divisors. The graph of $\frac{f_o(x)}{f(x)}$ is shown below.

Question: How many numbers $\le x$ have triangular divisors and why are even numbers more dense than odd numbers?

Related question: Reshaping an object into two integer sided cuboids without changing the total volume.