# Positive Integers which Divide Sum of All Lesser Primes

## Contents

## Theorem

The following sequence of positive integers have the property that each is a divisor of the sum of all primes smaller than them:

- $2, 5, 71, 369 \, 119, 415 \, 074 \, 643$

This sequence is A007506 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

As of time of writing (May 2017), no others are known.

## Proof

Verified by calculation.

## Examples

\(\displaystyle 2\) | \(=\) | \(\displaystyle 2 \times 0\) | There are no prime numbers less than $2$ |

\(\displaystyle 5\) | \(=\) | \(\displaystyle 2 + 3\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 1 \times 5\) |

\(\displaystyle 568\) | \(=\) | \(\displaystyle 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 8 \times 71\) |

## Historical Note

This sequence is reported by David Wells in *Curious and Interesting Numbers* as being noted by a contributor to *Journal of Recreational Mathematics* called **James Davies**, writing in Volume $14$.

An online listing of the articles for that journal has revealed an article titled *An Evening With Pi* by James Davis, in Volume $13$, but nothing specific for Volume $14$.

Direct investigation of the magazine in question will need to be done.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $71$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $369,119$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $71$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $369,119$