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A359137
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Primes such that there is a nontrivial permutation which when applied to the digits produces a prime (Version 2).
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6
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11, 13, 17, 31, 37, 71, 73, 79, 97, 101, 107, 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, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 419, 421
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OFFSET
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1,1
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COMMENTS
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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 not 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 A359136, we mean that the permutation itself is nontrivial.
There are only 53 primes not in this sequence, the greatest of which is 8059. - Andrew Howroyd, Jan 22 2023
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LINKS
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PROG
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(PARI) isok(n)={my(v=vecsort(digits(n))); if(#Set(v)<#v, 1, forperm(v, u, if(u[1]<>0, 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 p[0]!="0") if t!=n)
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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More than the usual number of terms are shown in order to distinguish this from other similar sequences.
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STATUS
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approved
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