A382127 -id:A382127 - 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

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

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

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

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

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

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

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

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.

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.