A376501 Primes that contain at least two different even digits where any permutation of the even digits leaving the odd digits fixed produces a prime. See comments for the treatment of 0.

241, 281, 283, 401, 421, 461, 463, 467, 601, 607, 641, 643, 647, 683, 809, 821, 823, 863, 1021, 1049, 1061, 1069, 1201, 1249, 1283, 1409, 1429, 1487, 1601, 1609, 1823, 1847, 2011, 2027, 2039, 2161, 2207, 2347, 2389, 2411, 2417, 2441, 2459, 2473, 2503, 2543, 2617, 2657, 2671, 2677, 2699, 2707

Offset: 1

Author

Enrique Navarrete, Sep 25 2024

Keywords

• nonn
• base

Comments

Primes for which permutations described in the name produce primes with leading 0s are in the sequence but the generated primes with leading 0s are not. For example, a transposition in 401 produces 041, hence 401 is in the sequence but 41 is not.

Examples

			2027, 2207 are primes and 227 is prime with a leading 0 generated by permuting even digits in either 2027 or 2207.  Hence 2027 and 2207 are in the sequence but 227 is not due to the leading 0.
6067, 6607 are primes but 667 generated by permuting even digits in either 6067 or 6607 is not prime, hence by name, neither number is in the sequence.
		

Links

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

Crossrefs

Cf. A000040, A003459, A376500, A376502.

Programs

Maple

filter:= proc(n) local L,oddi,eveni,xodd,i;
 if not isprime(n) then return false fi;
 L:= convert(n,base,10);
 oddi,eveni:= selectremove(t -> L[t]::odd,[$1..nops(L)]);
 if nops(convert(L[eveni],set))<2 then return false fi;
 xodd:= add(10^(i-1)*L[i],i=oddi);
 andmap(t -> isprime(xodd+add(10^(eveni[i]-1)*L[t[i]],i=1..nops(eveni))), combinat:-permute(eveni))
end proc:
select(filter, [seq(i,i=3..10000,2)]); # Robert Israel, Oct 23 2024