A108387 Doubly-transmutable primes: primes such that simultaneously exchanging pairwise all occurrences of any two disjoint pairs of distinct digits results in a prime.

113719, 131797, 139177, 139397, 193937, 313979, 317179, 317399, 331937, 371719, 739391, 779173, 793711, 793931, 797131, 917173, 971713, 971933, 979313, 997391, 1111793, 3333971, 7777139, 9999317, 13973731, 31791913, 79319197, 97137379

Offset: 1

Author

Rick L. Shepherd, Jun 02 2005

Keywords

• base
• nonn

Comments

By my definition of (a nontrivial) transmutable prime, each digit of each term must be capable of being an ending digit of a prime, so this sequence is a subsequence of A108387, primes p such that p's set of distinct digits is {1,3,7,9}. The repunit primes (A004022), which would otherwise trivially be (doubly-)transmutable and primes whose distinct digits are other proper subsets of {1,3,7,9} are excluded here by the two-disjoint-pair condition.

Examples

			a(0) = 113719 as this is the first prime having four distinct digits and such that all three simultaneous pairwise exchanges of all distinct digits as shown below 'transmutate' the original prime into other primes:
(1,3) and (7,9): 113719 ==> 331937 (prime),
(1,7) and (3,9): 113719 ==> 779173 (prime),
(1,9) and (3,7): 113719 ==> 997391 (prime).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A108387, A108388 (transmutable primes), A108389 (transmutable primes with four distinct digits), A107845 (transposable-digit primes), A003459 (absolute primes).

Programs

Maple

N:= 100: # to get a(1) to a(N)
R:= NULL: count:= 0:
S[1] := [0=1,1=3,2=7,3=9]:
S[2] := [0=3,1=1,2=9,3=7]:
S[3] := [0=7,1=9,2=1,3=3]:
S[4] := [0=9,1=7,2=3,3=1]:
g:= L -> add(L[i]*10^(i-1),i=1..nops(L)):
for d from 6 while count < N do
for n from 4^d to 2*4^d-1 while count < N do
  L:= convert(n,base,4)[1..-2];
  if nops(convert(L,set)) < 4 then next fi;
  if andmap(isprime,[seq(g(subs(S[i],L)),i=1..4)]) then
    R:= R, g(subs(S[1],L)); count:= count+1;
  fi
od od:
R; # Robert Israel, Jul 27 2020

Extensions

Offset changed by Robert Israel, Jul 27 2020