Numbers equal to Sum of Primes not Greater than its Prime Counting Function Value

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Theorem

Let $\map \pi n: \Z_{\ge 0} \to \Z_{\ge 0}$ denote the prime-counting function:

$\map \pi n =$ the count of the primes less than $n$

Consider the equation:

$\ds n = \sum_{p \mathop \le \map \pi n} p$

where $p \le \pi \left({n}\right)$ denotes the primes not greater than $\pi \left({n}\right)$.


Then $n$ is one of:

$5, 17, 41, 77, 100$

This sequence is A091864 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

We have that:

\(\ds \map \pi 5\) \(=\) \(\ds 3\)
\(\ds \leadsto \ \ \) \(\ds 2 + 3\) \(=\) \(\ds 5\)


\(\ds \map \pi {17}\) \(=\) \(\ds 7\)
\(\ds \leadsto \ \ \) \(\ds 2 + 3 + 5 + 7\) \(=\) \(\ds 17\)


\(\ds \map \pi {41}\) \(=\) \(\ds 13\)
\(\ds \leadsto \ \ \) \(\ds 2 + 3 + 5 + 7 + 11 + 13\) \(=\) \(\ds 41\)


\(\ds \map \pi {77}\) \(=\) \(\ds 21\)
\(\ds \leadsto \ \ \) \(\ds 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19\) \(=\) \(\ds 77\)


\(\ds \map \pi {100}\) \(=\) \(\ds 25\)
\(\ds \leadsto \ \ \) \(\ds 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23\) \(=\) \(\ds 100\)


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Sources

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