A225035 Primes such that there is a nontrivial rearrangement of the digits which is a prime.

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

Offset

1,1

Comments

The new prime is necessarily different from the original prime (so 11, for example) is not a term. - N. J. A. Sloane, Jan 22 2023

Permutations producing leading zeros are allowed: thus 101 is in the sequence because a nontrivial permutation of its digits is 011. - Robert Israel, Aug 13 2019

It seems reasonable to expect that the proportion of n-digit primes that are in this sequence approaches 1 as n increases. - Peter Munn, Sep 13 2022

References

H.-E. Richert, On permutation prime numbers, Norsk. Mat. Tidsskr. 33 (1951), p. 50-53.

Joe Roberts, Lure of the Integers, Math. Assoc. of Amer., 1992, p. 293.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Second Edition, Cambridge University Press, p. 121.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

13 is a term since a nontrivial permutation of its digits yields 31, which is also a prime.

Maple

dmax:=3: # for all terms of up to dmax digits
Res:= {}:
p:= 1:
do
  p:= nextprime(p);
  if p > 10^dmax then break fi;
  L:= sort(convert(p, base, 10), `>`);
  m:= add(L[i]*10^(i-1), i=1..nops(L));
  if assigned(A[m]) then
    if ilog10(A[m])=ilog10(p) then
      Res:= Res union {A[m], p}
    else Res:= Res union {p}
    fi
  else A[m]:= p
  fi
od:
sort(convert(Res, list)); # Robert Israel, Aug 13 2019

Mathematica

t={}; Do[p = Prime[n]; list1 = Permutations[IntegerDigits[p]]; If[Length[ Select[Table[FromDigits[n], {n, list1}], PrimeQ]] > 1, AppendTo[t, p]], {n, 84}]; t

Prog

(Python)
from sympy import isprime
from itertools import permutations
def ok(n):
    if not isprime(n): return False
    perms = (int("".join(p)) for p in permutations(str(n)))
    return any(isprime(t) for t in perms if t != n)
print([k for k in range(500) if ok(k)]) # Michael S. Branicky, Sep 14 2022
(PARI) is(p) = if(isprime(p), my(d=vecsort(digits(p))); d==vector(#d, x, 1)&&return(1); forperm(d, e, my(c = fromdigits(Vec(e))); p!=c && isprime(c) && return(1))); \\ Ruud H.G. van Tol, Jan 22 2023

Crossrefs

Cf. A052902, A007933, A007935, A166681.

See A055387, A359136-A359139 for other versions.

Sequence in context: A138375 A180526 A161401 * A344626 A006567 A263240

Adjacent sequences: A225032 A225033 A225034 * A225036 A225037 A225038

Keyword

nonn,base

Author

Jayanta Basu, Apr 24 2013

Extensions

Edited by N. J. A. Sloane, Jan 22 2023

Status

approved