A316787 Semipermutable Primes: One-digit primes and primes with 2 or more digits such that all permutations of their digits are primes except for permutations that place either 5 or even numbers in the units digit.

2, 3, 5, 7, 11, 13, 17, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 107, 113, 131, 149, 181, 199, 223, 227, 229, 241, 251, 277, 281, 283, 311, 337, 373, 401, 419, 421, 443, 449, 457, 461, 463, 467, 491, 503, 509, 521, 547, 557, 563, 569, 577, 587, 601, 607

Offset: 1

Author

Enrique Navarrete, Jul 13 2018

Keywords

• nonn
• base

Comments

Supersequence of A003459. The motivation of the sequence is to fill gaps in A003459.

The sequence contains all 1-digit primes, 20 2-digit primes (i.e., all 2-digit primes except 19), as opposed to only 9 2-digit primes in A003459, and 66 3-digit primes (as opposed to only 9 3-digit primes in A003459).

Also, the sequence contains 4-digit primes such as 4441 but also nontrivial ones such as 1181, 1811, 8111, which form an orbit of size 3 (see below), while there are no 4-digit primes in A003459.

If we call orbits the primes that can be obtained by such permutations, there are orbits of sizes 1,2,3, and 4 up to 3-digit primes.

In fact, there are only 3 orbits of size 4 up to 3-digit primes: {107, 17, 71, 701}, {149, 419, 491, 941} and {709, 79, 97, 907}.

It appears that there are no orbits of sizes larger than 4 for n-digit primes.

Permutations that have leading 0's are included: thus 409 is not in the sequence because 49 is not prime. - Robert Israel, Aug 31 2018

Examples

			127 is not in the sequence since 271 is prime but neither 217 nor 721 are; to be in the sequence all of these numbers would have to be prime, and they would form an orbit of size 4 (by Name, permutations of these numbers ending in 2 are not considered).
241 and 421 are in the sequence and form an orbit of size 2 since these primes can be obtained by permutations that forbid the units digit to be an even number.
569 and 659 are in the sequence since these primes can be obtained by permutations that forbid the units digit to be either 5 or an even number.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..134

Crossrefs

Contains A003459, A105976, A105978, A105979, A105980, A105982.

Programs

Maple

filter:= proc(n) local L,m,i,t;
  if not isprime(n) then return false fi;
  L:= convert(n,base,10);
  m:=nops(L);
  for i in select(t -> member(L[t],[1,3,7,9]), [$1..m]) do
    for t in combinat:-permute(subsop(i=NULL, L)) do
      if not isprime(L[i]+add(10^j*t[j],j=1..m-1)) then
        return false fi
  od od;
  true
end proc:
select(filter, [2,seq(i,i=3..2000,2)]); # Robert Israel, Aug 31 2018

Mathematica

Select[Prime@Range[120], AllTrue[FromDigits /@ Permutations[IntegerDigits@ #], PrimeQ[#] || MemberQ[{0, 2, 4, 5, 6, 8}, Mod[#, 10]] &] &] (* Giovanni Resta, Jul 14 2018 *)