Weakly prime number

In number theory, a prime number is called weakly prime if it becomes composite when any one of its digits is changed to every single other digit. Decimal digits are usually assumed.

The first weakly prime numbers are:

294001, 505447, 584141, 604171, 971767, 1062599, 1282529, 1524181, 2017963, 2474431, 2690201, 3085553, 3326489, 4393139, ... (sequence A050249 in the OEIS)

For the first of these, each of the 54 numbers 094001, 194001, 394001, ..., 294009 are composite. A weakly prime base-b number with n digits must produce (b−1) × n composite numbers when a digit is changed.

In 2007 Jens Kruse Andersen found the 1000-digit weakly prime (17×10¹⁰⁰⁰−17)/99 + 21686652. This is the largest known weakly prime number as of 2011.

There are infinitely many weakly prime numbers in any base. Furthermore, for any fixed base there is a positive proportion of such primes.

The smallest base b weakly primes for b = 1 to 16 are: (sequence A186995 in the OEIS)

11₁ = 2 1111111₂ = 127 2₃ = 2 11311₄ = 373 313₅ = 83 334155₆ = 28151 436₇ = 223 14103₈ = 6211 3738₉ = 2789 294001₁₀ = 294001 2573₁₁ = 3347 6B8AB77₁₂ = 20837899 2216₁₃ = 4751 C371CD₁₄ = 6588721 9880C₁₅ = 484439 D2A45₁₆ = 862789