Positive Integers which Divide Sum of All Lesser Primes

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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