Is there any other number that has similar properties as $21$?

Solution 1:

This is a very interesting question…

$\newcommand{sopfr}{\operatorname{sopfr}}$ $f(n)=\frac{n(n+1)}2$ and $g(n)=\sopfr(n)$, the sum of prime factors of $n$ with repeats (OEIS A001414). We want $n$ such that $g(f(n))=n$ or $$\sopfr\left(\frac{n(n+1)}2\right)=n\tag1$$ which can be split into two cases due to the property $\sopfr(ab)=\sopfr(a)+\sopfr(b)$.

• If $n$ is even, then $\sopfr\left(\frac n2\right)+\sopfr(n+1)=n$. We know that $\sopfr(n)\le n$, so $\sopfr\left(\frac n2\right)\le\frac n2$ and consequently $\sopfr(n+1)\ge\frac n2$. Either $n+1$ is a prime, in which case the LHS of $(1)$ is greater than $n$ and so the equality cannot hold, or $n+1$ is odd composite and so has a least prime factor at least 3*, yielding $\sopfr(n+1)\le3+\frac{n+1}3$ and thus $$\frac n2\le3+\frac{n+1}3$$ which is only true for $n\le20$. Checking those $n$ reveals no solutions to $(1)$.
• If $n$ is odd, the reasoning is similar: $\sopfr\left(\frac{n+1}2\right)+\sopfr(n)=n$, where $\sopfr\left(\frac{n+1}2\right)\le\frac{n+1}2$ and so $\sopfr(n)\ge\frac{n-1}2$. Since $n$ is odd, either it is prime and the LHS of $(1)$ is greater than $n$, or it has a least prime factor at least 3* and $\sopfr(n)\le3+\frac n3$, giving $$\frac{n-1}2\le3+\frac n3$$ which only holds for $n\le21$. 21 is the solution to $(1)$ pointed out in the original question; we have just shown it is the only one.

*Technically we have to repeat the argument for other possible least prime factors $k$ of $n$ or $n+1$ – and the upper bound $N_k$ of the solution to the inequalities in $n$ increases accordingly, each 3 replaced with $k$. However, the least composite number with least prime factor $k$ is $k^2$, and this increases much faster than $N_k$ (which is $\sim\frac k2$). Indeed, $5^2$ already exceeds $N_5$ for both inequalities.

The method I use above has very strong similarities to the method I used in my most famous answer of all. It is sheer coincidence that 21 is a solution to both the problems I answered.