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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. 3
13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 179, 191, 199, 313, 331, 337, 773, 911, 919, 1171, 1933, 3391, 7717, 9311, 11113, 11119, 11177, 11717, 11933, 33199, 33331, 77171, 77711, 77713, 79999, 97777, 99991, 113111, 131111, 131113, 131171, 131311 (list; graph; refs; listen; history; text; internal format)



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).


Table of n, a(n) for n=0..41.


179 is a term because it is prime and its three transmutations are all prime:

exchanging ('transmuting') 1 and 7: 179 ==> 719 (prime),

exchanging 1 and 9: 179 ==> 971 (prime) and

exchanging 7 and 9: 179 ==> 197 (prime).

(As 791 and 917 are not prime, 179 is not a term of A068652 or A003459 also.).

Similarly, 1317713 is transmutable:

exchanging all 1's and 3s: 1317713 ==> 3137731 (prime),

exchanging all 1's and 7s: 1317713 ==> 7371173 (prime) and

exchanging all 3s and 7s: 1317713 ==> 1713317 (prime).


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).

Sequence in context: A263240 A246043 A246045 * A083983 A129338 A138535

Adjacent sequences:  A108385 A108386 A108387 * A108389 A108390 A108391




Rick L. Shepherd, Jun 02 2005



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Last modified November 27 18:27 EST 2015. Contains 264550 sequences.