A109093
Fully-transmutable primes: Transmutable primes such that each transmutation is itself a transmutable prime (A108388).
Original entry on oeis.org
139119131, 193113191, 319339313, 391331393, 913993919, 931991939, 1319999199391, 1913333133931, 3139999399193, 3931111311913, 9193333933139, 9391111911319, 11333911193113, 11999311139119, 33111933391331
Offset: 0
The first six terms share the digit pattern d1 d2 d3 d1 d1 d3 d1 d2 d1. Each of these terms is a (9-digit) prime corresponding to one of the 3! = 6 bijective mappings of {1,3,9} onto {d1,d2,d3}. There are no other such primes with nine or fewer digits.
Cf.
A108388 (transmutable primes),
A083983 (transmutable primes with two distinct digits),
A108389 (transmutable primes with four distinct digits),
A003459 (absolute primes),
A108387 (doubly-transmutable primes).
A108387
Doubly-transmutable primes: primes such that simultaneously exchanging pairwise all occurrences of any two disjoint pairs of distinct digits results in a prime.
Original entry on oeis.org
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
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).
-
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
A108389
Transmutable primes with four distinct digits.
Original entry on oeis.org
133999337137, 139779933779, 173139331177, 173399913979, 177793993177, 179993739971, 391331737931, 771319973999, 917377131371, 933971311913, 997331911711, 1191777377177, 9311933973733, 9979333919939, 19979113377173, 31997131171111, 37137197179931, 37337319113911
Offset: 1
a(0)=133999337137 is the smallest transmutable prime with four distinct digits (1,3,7,9):
exchanging all 1's and 3's: 133999337137 ==> 311999117317 (prime),
exchanging all 1's and 7's: 133999337137 ==> 733999331731 (prime),
exchanging all 1's and 9's: 133999337137 ==> 933111337937 (prime),
exchanging all 3's and 7's: 133999337137 ==> 177999773173 (prime),
exchanging all 3's and 9's: 133999337137 ==> 199333997197 (prime) and
exchanging all 7's and 9's: 133999337137 ==> 133777339139 (prime).
No smaller prime with four distinct digits transmutes into six other primes.
Cf.
A108386 (Primes p such that p's set of distinct digits is {1, 3, 7, 9}),
A108388 (transmutable primes),
A083983 (transmutable primes with two distinct digits),
A108387 (doubly-transmutable primes),
A006567 (reversible primes),
A002385 (palindromic primes),
A068652 (every cyclic permutation is prime),
A107845 (transposable-digit primes),
A003459 (absolute primes),
A057876 (droppable-digit primes).
Showing 1-3 of 3 results.
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