%I A108388
%S A108388 13,17,31,37,71,73,79,97,113,131,179,191,199,313,331,337,773,911,919,
%T A108388 1171,1933,3391,7717,9311,11113,11119,11177,11717,11933,33199,33331,
%U A108388 77171,77711,77713,79999,97777,99991,113111,131111,131113,131171,131311
%N A108388 Transmutable primes: Primes with distinct digits d_i, i=1,m (2<=m<=4) 
               such that simultaneously exchanging all occurrences of any one pair 
               (d_i,d_j), i<>j results in a prime.
%C A108388 a(n) is a term iff a(n) is prime and binomial(m,2) 'transmutations' (see 
               example) of a(n) are different primes. A083983 is the subsequence 
               for m=2: one transmutation (The author of A083983, Amarnath Murthy, 
               calls the result of such a digit-exchange a self-complement. {Because 
               I didn't know until afterwards that this sequence was a generalization 
               of A083983 and as this generalization always leaves some digits unchanged 
               for m>2, I've chosen different terminology.}). A108389 ({1,3,7,9}) 
               is the subsequence for m=4: six transmutations. Each a(n) corresponding 
               to m=3 (depending upon its set of distinct digits) and having three 
               transmutations is also a member of A108382 ({1,3,7}), A108383 ({1,
               3,9}), A108384 ({1,7,9}), or A108385 ({3,7,9}). The condition m>=2 
               only eliminates the repunit (A004022) and single-digit primes. The 
               condition m<=4 is not a restriction because if there were more distinct 
               digits, they would include even digits or the digit 5, in either 
               case transmuting into a composite number. Some terms such as 1933 
               are reversible primes ("Emirps": A006567) and the reverse is also 
               transmutable. The transmutable prime 3391933 has three distinct digits 
               and is also a palindromic prime (A002385). The smallest transmutable 
               prime having four distinct digits is A108389(0) = 133999337137 (12 
               digits).
%e A108388 179 is a term because it is prime and its three transmutations are all 
               prime:
%e A108388 exchanging ('transmuting') 1 and 7: 179 ==> 719 (prime),
%e A108388 exchanging 1 and 9: 179 ==> 971 (prime) and
%e A108388 exchanging 7 and 9: 179 ==> 197 (prime).
%e A108388 (As 791 and 917 are not prime, 179 is not a term of A068652 or A003459 
               also.).
%e A108388 Similarly, 1317713 is transmutable:
%e A108388 exchanging all 1's and 3s: 1317713 ==> 3137731 (prime),
%e A108388 exchanging all 1's and 7s: 1317713 ==> 7371173 (prime) and
%e A108388 exchanging all 3s and 7s: 1317713 ==> 1713317 (prime).
%Y A108388 Cf. A108382, A108383, A108384, A108385, A108386, A108389 (transmutable 
               primes with four distinct digits), A083983 (transmutable primes with 
               two distinct digits), A108387 (doubly-transmutable primes), A006567 
               (reversible primes), A002385 (palindromic primes), A068652 (every 
               cyclic permutation is prime), A003459 (absolute primes).
%Y A108388 Sequence in context: A138375 A161401 A006567 this_sequence A083983 A129338 
               A138535
%Y A108388 Adjacent sequences: A108385 A108386 A108387 this_sequence A108389 A108390 
               A108391
%K A108388 base,nonn
%O A108388 0,1
%A A108388 Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jun 02 2005