

A316787


Semipermutable Primes: Onedigit 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.


1



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
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OFFSET

1,1


COMMENTS

Supersequence of A003459. The motivation of the sequence is to fill gaps in A003459.
The sequence contains all 1digit primes, 20 2digit primes (i.e., all 2digit primes except 19), as opposed to only 9 2digit primes in A003459, and 66 3digit primes (as opposed to only 9 3digit primes in A003459).
Also, the sequence contains 4digit 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 4digit 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 3digit primes.
In fact, there are only 3 orbits of size 4 up to 3digit 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 ndigit 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


LINKS

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


EXAMPLE

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.


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..m1)) 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 *)


CROSSREFS

Contains A003459, A105976, A105978, A105979, A105980, A105982.
Sequence in context: A290959 A003309 A063884 * A165671 A162855 A194954
Adjacent sequences: A316784 A316785 A316786 * A316788 A316789 A316790


KEYWORD

nonn,base


AUTHOR

Enrique Navarrete, Jul 13 2018


STATUS

approved



