A382198 -id:A382198 - cp's OEIS frontend

A382179 Numbers k such that for each digit of k, 2k(digit) + 1 is prime.

1, 3, 6, 9, 11, 14, 15, 22, 24, 25, 27, 28, 33, 44, 54, 63, 75, 78, 81, 88, 99, 111, 119, 131, 141, 153, 168, 173, 219, 249, 252, 255, 279, 282, 322, 325, 333, 357, 363, 414, 441, 459, 474, 491, 538, 553, 558, 565, 611, 666, 674, 699, 794, 797, 828, 831, 832, 858, 895, 924, 947, 955

Offset: 1

Jakub Buczak, Mar 17 2025

The decision to use the expression 2*k*(digit) instead of k*(digit) is based on the fact that k*(odd) + 1 is never prime. By multiplying k*(digit) by 2, we ensure that any number, regardless of its digits, can appear in the sequence. Additionally, numbers containing the digit 0 are never terms, as 2*k*(0) + 1 is never prime.
When k is restricted to the smallest term with n distinct digits, only 7 terms exist (see A382198).
If the smallest term k is further restricted to a prime number p with n distinct digits, the conditions become significantly more restrictive (see A382127).

			63 is a term because 2*63*6 + 1 and 2*63*3 + 1 are both prime.

Jakub Buczak, Table of n, a(n) for n = 1..4000

Cf. A000040, A382127, A382198, A382199.

Select[Range[2500], AllTrue[2 * # * IntegerDigits[#] + 1, PrimeQ] &] (* Amiram Eldar, Mar 17 2025 *)

isok(k) = my(d=Set(digits(k))); for (i=1, #d, if (!isprime(2*k*d[i]+1), return(0));); return(1); \\ Michel Marcus, Mar 17 2025

from sympy import isprime
def ok(n): return all(isprime(2*n*d+1) for d in map(int, set(str(n))))
print([k for k in range(956) if ok(k)]) # Michael S. Branicky, Mar 17 2025

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

Original entry on oeis.org

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

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

A382179 Numbers k such that for each digit of k, 2k(digit) + 1 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Python

Extensions

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

Views

Author

Keywords

Crossrefs

Programs

PARI

cp's OEIS Frontend

A382179 Numbers k such that for each digit of k, 2*k*(digit) + 1 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Python

Extensions

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

Views

Author

Keywords

Crossrefs

Programs

PARI

A382179 Numbers k such that for each digit of k, 2k(digit) + 1 is prime.

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

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