

A108387


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


4



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


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 twodisjointpair condition.


LINKS



EXAMPLE

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).


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^(i1), i=1..nops(L)):
for d from 6 while count < N do
for n from 4^d to 2*4^d1 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:


CROSSREFS



KEYWORD

base,nonn


AUTHOR



EXTENSIONS



STATUS

approved



