# How often can a number be written as a linear combination of the squares of its prime divisors?

Peter asked here "Can a number be equal to the sum of the squares of its prime divisors?" and, it seems clear that if $$n=p_1^{a_1}\cdots p_k^{a_k},$$ and $$f(n):=p_1^2+\cdots+p_k^2$$ that then $$n=f(n)$$ only if $$n$$ is a square of a prime.

I almost proved that, but did not yet find so good and simple bounds to show that indeed it is possible only if $$n$$ is a square of a prime.

But, despite the fact I still do not have a proof, I decided to ask a little more general question:

If $$n=p_1^{a_1}\cdots p_k^{a_k},$$ is such that $$n=b_1p_1^2+...+b_kp_k^2$$ for some $$b_1,...b_k \in \mathbb N$$, then what is the density of the set of all such $$n$$ in $$\mathbb N$$?

I am also interested in how often do you find more than the one set $$(b_1,...b_k)$$ that solves $$p_1^{a_1}...p_k^{a_k}=b_1p_1^2+...+b_kp_k^2$$? That is, how often a function $$n \to (b_1,...b_k)$$ is multi-valued?

What is the biggest $$n$$ for which you did not find a solution?

Is the sequence of $$n$$´s for which there is no solution in OEIS?

• For small values of $n$, it shouldn't be hard for you to work out whether or not there is a solution. Then you can see for yourself whether that sequence is in the OEIS. Why not do that? – Gerry Myerson Jun 23 at 10:23
• @GerryMyerson I am not skilled in programming, almost all calculations and proofs that I do are old-school fashioned, they are written on paper. Yes, I can see some small $n$´s that are not solutions. – Grešnik Jun 23 at 10:30
• So, it's easy to see that if $n$ is prime there's no solution, right? That doesn't take any programming. If $n$ is the square of a prime, or any higher power of a prime, there is a solution, right? Now, $6=2\times3$, and there's no way to write $6$ as $4a+9b$. So at this point we know the sequence starts $2,3,5,6,7$ but not $8$ or $9$. It shouldn't be hard for you to do a few more cases without any programming, and then look up the results. Please, do it! – Gerry Myerson Jun 23 at 12:37
• @GerryMyerson With only "by heart" calculations it seems to me that there are also no solutions for n=10,11,12,14,15,16..and so on. Yes, there are solutions for some prime n, why did you think there are not? – Grešnik Jun 23 at 12:53
• How do you write $13$ as a linear combination of $13^2$? – Gerry Myerson Jun 23 at 12:55