A382127 Smallest prime p with n distinct digits, such that for each digit of p, 2*p*(digit) + 1 is prime.

3, 131, 173, 4391, 4746616799

Offset: 1

Author

Jakub Buczak, Mar 16 2025

Keywords

• nonn
• base
• fini
• full

Comments

The sequence is proven to be finite by definition and by nature, containing only up to 7 terms (though it is uncertain if all 7 exist). This is because in base 10 there are 10 digits, excluding 0 that's 9. There are always 2 digits d_1 and d_2, such that 2*p*d_1 + 1 and 2*p*d_2 + 1 has an ending digit of 5. The (d_1,d_2) for the ending digits of p are: 1->(2,7), 3->(4,9), 7->(1,6), 9->(3,8). We exclude any prime with a digit 0, because 2*p*(0) + 1 is never prime.

The form 2*p*(digit) + 1 ensures a chance of primality, as p*(odd) + 1 is always composite.

Under 2*p*(digit) + 1, every term with a digit 1 is also a Sophie-Germain prime (see A005384).

Examples

			a(2) = 131, because 131 has exactly 2 distinct digits (1,3), and 2*131*1 + 1 and 2*131*3 + 1 are both prime.
		

Crossrefs

Subsequence of A382199.

Cf. A000040, A005384, A382179, A382198.

Programs

PARI

isok(k, n) = my(d=Set(digits(k))); if (#d != n, return(0)); for (i=1, #d, if (!isprime(2*k*d[i]+1), return(0)); ); return(1);
a(n) = my(p=2); while (!isok(p, n), p=nextprime(p+1)); p; \\ Michel Marcus, Mar 18 2025

Python

from sympy import isprime
from itertools import count
def a(n):
    return next(p for p in count(2) if isprime(p) and len(set(str(p)))==n and '0' not in str(p) and all(isprime(2*p*int(d)+1) for d in set(str(p))))

Extensions

a(5) from Michel Marcus, Mar 18 2025