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A108387 Doubly-transmutable 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 two-disjoint-pair condition.

LINKS

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

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^(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

CROSSREFS

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

Sequence in context: A256901 A066790 A135411 * A112009 A034633 A253118

Adjacent sequences:  A108384 A108385 A108386 * A108388 A108389 A108390

KEYWORD

base,nonn

AUTHOR

Rick L. Shepherd, Jun 02 2005

EXTENSIONS

Offset changed by Robert Israel, Jul 27 2020

STATUS

approved

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Last modified August 13 07:13 EDT 2022. Contains 356078 sequences. (Running on oeis4.)