Prime Numbers which Divide Sum of All Lesser Primes
Jump to navigation
Jump to search
Theorem
The following sequence of prime numbers has 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 (April $2020$), no others are known.
Proof
Verified by calculation.
Examples
| \(\ds 2\) | \(=\) | \(\ds 2 \times 0\) | There are no prime numbers less than $2$ |
| \(\ds 5\) | \(=\) | \(\ds 2 + 3\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1 \times 5\) |
| \(\ds 568\) | \(=\) | \(\ds 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 8 \times 71\) |
Sources
- 1981 -- 82: Jeffrey Shallit: Prime Sums (J. Recr. Math. Vol. 14, no. 1: p. 44)
- 1982: H.L. Nelson: Prime Sums (J. Recr. Math. Vol. 14, no. 3: p. 205)
- 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): $369,119$