A272022 Look at the set of numbers obtained by permuting the digits of n in all possible ways, then remove n itself from the set. If the remaining numbers are all primes, then n is in the sequence.

13, 14, 16, 17, 20, 30, 31, 32, 34, 35, 37, 38, 50, 70, 71, 73, 74, 76, 79, 91, 92, 95, 97, 98, 110, 113, 118, 119, 131, 133, 199, 311, 337, 373, 733, 772, 775, 778, 779, 919, 991, 1118, 3337, 7771, 77779

Offset: 1

Author

César Eliud Lozada, Apr 18 2016

Keywords

• nonn
• base
• more

Comments

If it exists, a(46) > 5*10^11. - Lars Blomberg, Mar 31 2018

Examples

			119 is in the sequence because every permutation of its digits excluding 119 (i.e., 191 and 911) is a prime.
11 is not in the sequence, because when 11 is removed from the set, no numbers are left.
		

Crossrefs

Cf. A095179, A111347.

Cf. A003459. - Altug Alkan, Apr 18 2016

Programs

Maple

lis := [];
for n from 1 to 10000 do
  nn := convert(n, base, 10);
  pp := combinat[permute](nn);
  if nops(pp) = 1 then
    next
  end if;
  lOk := true;
  for p in pp do
    if p = nn then
      next: #exclude n
    end if;
    if `not`(isprime(convert(p, base, 10, 10^nops(p))[])) then
      lOk := false; break
    end if
  end do;
  if lOk then
    lis := [op(lis), n]
  end if
end do:
lis := lis;

Mathematica

rnapQ[n_]:=Module[{p=Rest[FromDigits/@Permutations[IntegerDigits[ n]]]},If[ Length[p]==0, False, AllTrue[p,PrimeQ]]]; Select[Range[80000],rnapQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 24 2019 *)

PARI

isok(n) = {v = []; d = digits(n); for (k=0, (#d)!-1, p = numtoperm(#d, k); dp = vector(#d, j, d[p[j]]); np = subst(Pol(dp), x, 10); v = Set(concat(v, np));); v = setminus(v, Set(n)); if (#v == 0, return (0)); for (k=1, #v, if (!isprime(v[k]), return (0));); return (1);} \\ Michel Marcus, Apr 18 2016

Python

from sympy import isprime
from itertools import count, islice, permutations
def agen(): yield from (k for k in count(1) if len(set(s:=str(k)))!=1 and all((t:=int("".join(m)))==k or isprime(t) for m in permutations(s)))
print(list(islice(agen(), 45))) # Michael S. Branicky, Dec 29 2023