Primeval number
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In mathematics, a primeval number is a natural number n for which the number of prime numbers which can be obtained by permuting all or some of its digits (in base 10) is larger than the number of primes obtainable in the same way for any smaller natural number. Primeval numbers were first described by Mike Keith.
The first few primeval numbers are 1, 2, 13, 37, 107, 113, 137, 1013, 1037, 1079, 1237, 1367, ... (sequence A072857 in OEIS); the number of primes that can be obtained from the primeval numbers is 0, 1, 3, 4, 5, 7, 11, 14, 19, 21, 26, 29, ... (sequence A076497 in OEIS). The number of primes that can be obtained from a primeval number with n digits is 1, 4, 11, 31, 106, ... (sequence A076730 in OEIS).
An example of how this works, matching the first five primeval numbers with the number of primes obtained is given:
| Primeval number | Primes obtained | Number of primes contained |
|---|---|---|
| (sequence A072857 in OEIS) | (ordered permutations) | (sequence A088130 in OEIS) |
| 2 | 2 | 1 |
| 13 | 3, 13, 31 | 3 |
| 37 | 3, 7, 37, 73 | 4 |
| 107 | 7, 17, 71, 107, 701 | 5 |
| 113 | 3, 11, 13, 31, 113, 131, 311 | 7 |
