6

I was messing around with the prime counting function $\pi(n)$ because I was bored, but then I noticed something.

The equation $$\pi\left(\sum_{i=1}^{n}\pi(i)\right)-1=n,$$ has the solution $n=13$. Are there any numbers like $13$ that satisfy this equation? Is there a way of finding like values? Has this been discovered before?

If it can be proven that $13$ is the only number (let alone prime) with this property, I will feel like Ulam after discovering upon the famous prime spiral. It is otherwise my conjecture as I am unaware of anyone else finding out this property.

Klangen
  • 5,075
Mr Pie
  • 9,459
  • 4
    Other solutions are $n=14, 15$ – gammatester Apr 09 '18 at 10:11
  • @gammatester and that's why I should always test some more myself. I was just bored and a bit lazy today... Is it known whether or not there are infinitely many numbers like this? – Mr Pie Apr 09 '18 at 10:15
  • 1
    I do not believe so. The function is roughly monotone increasing (with a few exceptions). I guess the bounds for $\pi(n)$ can be used to disprove your conjecture. – gammatester Apr 09 '18 at 10:21
  • 3
    I believe (if I understand well your formula, I don't know why is tagged with the tag (pi)) that is better write your formula/equation/condition as $\pi\left(\sum_{k=1}^n\pi(k)\right)=n+1$, where $\pi(x)$ is the prime-counting function. I add also as companion of previous comments a line code written in Pari-GP (search in Internet Sage Cell Server and choose GP as language) for (i = 1, 1000,if(primepi(sum( k=1, i, primepi(k)))==i+1,print(i))) –  Apr 09 '18 at 10:31
  • "If $\ldots$ then $n=13$ satisfies." Could you fix the syntax because this makes no sense right now. On the left is a number, on the right is a set. @gammatester what did you understand of this? – Arnaud Mortier Apr 09 '18 at 10:37
  • @ArnaudMortier I use the set notation because I have a "such that" in the equation, and thus allows various values of $n$. This is actually very common notation. I will change the syntax though. – Mr Pie Apr 09 '18 at 10:44
  • 4
    @arnaud-mortier: I considered the zeros of the function $$f(n) = \pi\Big(\sum_{i=1}^n \pi(i)\Big)-n-1$$ – gammatester Apr 09 '18 at 10:51
  • 1
    @user243301 yes you have understood it correctly. – Mr Pie Apr 09 '18 at 10:53
  • Thanks for your response, that I mean is the same thing that Arnaud. My emphasis, like that of other users, is to be able to write your nice question, the title and the body rigorously. –  Apr 09 '18 at 10:56
  • 3
    @gammatester Thanks. (at)user477343 No, this is not a common notation. A number and a set are two different things unless you are in pure logic which is not the case here, and you do not define a function inside a set. – Arnaud Mortier Apr 09 '18 at 11:06
  • @ArnaudMortier I was taught to use this notation from a maths book. I forgot what it's called but it was written by a famous author and talked about functions, sets, probability, limits, and some trig and other parts of calculus, also radicals. I guess that's every maths book... but yeah. I thought this was normal notation. – Mr Pie Apr 09 '18 at 11:13
  • @Servaes thank you for the edit! My phone died before I could edit it, hahah. It's charging now so all good :) – Mr Pie Apr 09 '18 at 11:24
  • 6
    The LHS grows as $\left(\frac{n}{\ln{n}}\right)^2$, so faster than $n$. So find some lower bounds for $\pi(k)$ and use them to find an upper bound on $n$ for which the equality could hold; probably you'll find some $n<100$. Then it is a matter of checking these few cases by hand or computer. – Servaes Apr 09 '18 at 12:05
  • 2
    Checked by computer, there are no other solutions except $n=13, 14, 15$, all the way up to $n=10^4$. And the value of the expression on the left side grows faster than $n$. There are no other solutions. – Saša Apr 09 '18 at 12:49
  • @Oldboy thank you very much. I went to the following links and they helped: https://math.stackexchange.com/questions/1890741/what-is-the-simplest-lower-bound-on-prime-counting-functions-proof-wise/1890792 and https://math.stackexchange.com/questions/59258/lower-bound-for-the-prime-number-function – Mr Pie Apr 11 '18 at 05:47

0 Answers0