A359136 Primes such that there is a nontrivial permutation which when applied to the digits produces a prime (Version 1).

11, 13, 17, 31, 37, 71, 73, 79, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 419, 421

Offset: 1

Author

N. J. A. Sloane and Allan C. Wechsler, Jan 22 2023

Keywords

• nonn
• base

Comments

A prime p with decimal expansion p = d_1 d_2 ... d_m is in this sequence iff there is a non-identity permutation pi in S_m such that q = d_pi(1) d_pi(2) ... d_pi(m) is also a prime. The prime q may or may not be equal to p. Leading zeros are permitted in q.

One must be careful when using the phrase "nontrivial permutation of the digits". When the first and third digits of 101 are exchanged, this is applying the nontrivial permutation (1,3) in S_3 to the digits, leaving the number itself unchanged. One should specify whether it is the permutation that is nontrivial, or its action on what is being permuted. In this sequence and A359137, we mean that the permutation itself is nontrivial.

There are only 34 primes not in this sequence, the greatest of which is 5849. - Andrew Howroyd, Jan 22 2023

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000

Crossrefs

Many OEIS entries are subsequences (possibly after omitting 2, 3, 5, and 7): A007500, A055387, A061461, A069706, A090933, A225035.

Cf. A359137, A359138, A359139.

Programs

PARI

isok(n)={my(v=vecsort(digits(n))); if(#Set(v)<#v, 1, forperm(v, u, my(t=fromdigits(Vec(u))); if(isprime(t) && t<>n, return(1))); 0)} \\ Andrew Howroyd, Jan 22 2023

Python

from sympy import isprime
from itertools import permutations as P
def ok(n):
    if not isprime(n): return False
    if len(s:=str(n)) > len(set(s)): return True
    return any(isprime(t) for t in (int("".join(p)) for p in P(s)) if t!=n)
print([k for k in range(422) if ok(k)]) # Michael S. Branicky, Jan 23 2023

Extensions

More than the usual number of terms are shown in order to distinguish this from neighboring sequences.

Incorrect terms removed by Andrew Howroyd, Jan 22 2023