A382179 -id:A382179 - cp's OEIS frontend

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

3, 131, 173, 4391, 4746616799

Offset: 1

Jakub Buczak, Mar 16 2025

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).

			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.

Subsequence of A382199.

Cf. A000040, A005384, A382179, A382198.

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

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))))

a(5) from Michel Marcus, Mar 18 2025

A382198 Smallest integer k with n distinct digits, such that for each digit of k, 2k(digit) + 1 is prime.

Original entry on oeis.org

3, 14, 153, 2169, 48165, 125769, 327174495

Offset: 1

Michel Marcus, Mar 18 2025

See 1st comment of A382127 for an explanation that there cannot be more than 7 terms.

Subsequence of A382179.

Cf. A382127, A382199.

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(k=2); while (!isok(k, n), k++); k;

A382199 Primes p such that for each digit of p, 2p(digit) + 1 is prime.

Original entry on oeis.org

3, 11, 131, 173, 491, 797, 947, 1931, 3583, 4391, 6173, 7937, 32323, 49919, 64499, 79997, 83383, 149111, 232333, 296269, 366161, 477947, 611333, 616169, 616961, 635563, 667673, 969179, 1111991, 1779779, 2232523, 2662669, 2922229, 3444341, 5333353, 5599999, 6853663, 6919691, 6929929

Offset: 1

Michel Marcus, Mar 18 2025

Subsequence of primes of A382179.

Cf. A382127, A382198.

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

cp's OEIS Frontend

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

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A382198 Smallest integer k with n distinct digits, such that for each digit of k, 2k(digit) + 1 is prime.

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A382199 Primes p such that for each digit of p, 2p(digit) + 1 is prime.

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cp's OEIS Frontend

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

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A382198 Smallest integer k with n distinct digits, such that for each digit of k, 2*k*(digit) + 1 is prime.

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PARI

A382199 Primes p such that for each digit of p, 2*p*(digit) + 1 is prime.

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PARI

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

A382198 Smallest integer k with n distinct digits, such that for each digit of k, 2k(digit) + 1 is prime.

A382199 Primes p such that for each digit of p, 2p(digit) + 1 is prime.