Primeval number

In mathematics, a primeval number is a natural number n for which the number of prime numbers which can be obtained by permuting some or all 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, 1379, 10079, 10123, 10136, 10139, 10237, 10279, 10367, 10379, 12379, 13679, ... (sequence A072857 in the 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, 31, 33, 35, 41, 53, 55, 60, 64, 89, 96, 106, ... (sequence A076497 in the OEIS)

The largest number of primes that can be obtained from a primeval number with n digits is

1, 4, 11, 31, 106, 402, 1953, 10542, 64905, 362451, 2970505, ... (sequence A076730 in the OEIS)

The smallest n-digit number to achieve this number of primes is

2, 37, 137, 1379, 13679, 123479, 1234679, 12345679, 102345679, 1123456789, 10123456789, ... (sequence A134596 in the OEIS)

Primeval numbers can be composite. The first is 1037 = 17×61. A Primeval prime is a primeval number which is also a prime number:

2, 13, 37, 107, 113, 137, 1013, 1237, 1367, 10079, 10139, 12379, 13679, 100279, 100379, 123479, 1001237, 1002347, 1003679, 1012379, ... (sequence A119535 in the OEIS)

The following table shows the first seven primeval numbers with the obtainable primes and the number of them.

Primeval numberPrimes obtainedNumber of primes
10
221
133, 13, 313
373, 7, 37, 734
1077, 17, 71, 107, 7015
1133, 11, 13, 31, 113, 131, 3117
1373, 7, 13, 17, 31, 37, 71, 73, 137, 173, 31711

In base 12, the primeval numbers are: (using inverted two and three for ten and eleven, respectively)

1, 2, 13, 15, 57, 115, 117, 125, 135, 157, 1017, 1057, 1157, 1257, 125Ɛ, 157Ɛ, 167Ɛ, ...

The number of primes that can be obtained from the primeval numbers is: (written in base 10)

0, 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 20, 23, 27, 29, 33, 35, ...
Primeval numberPrimes obtainedNumber of primes (written in base 10)
10
221
133, 312
155, 15, 513
575, 7, 57, 754
1155, 11, 15, 51, 5115
1177, 11, 17, 117, 171, 7116
1252, 5, 15, 25, 51, 125, 2517
1353, 5, 15, 31, 35, 51, 315, 5318
1575, 7, 15, 17, 51, 57, 75, 157, 175, 517, 75111

Note that 13, 115 and 135 are composite: 13 = 3×5, 115 = 7×1Ɛ, and 135 = 5×31.

See also

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