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
- 1991: Solomon W. Golomb and Marcin E. Kuczma: A Perfect Property of Handful of 2 Numbers: Problem E3385 (Amer. Math. Monthly Vol. 98, no. 9: pp. 858 – 859) www.jstor.org/stable/2324281
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $100$