OFFSET
1,1
COMMENTS
Smaller members of Ormiston prime pairs.
Given the n-th prime, it is occasionally possible to form the (n+1)th prime using the same digits in a different order. Such a pair is called an Ormiston pair.
Ormiston pairs occur rarely but randomly. It is thought that there are infinitely many but this has not been proved. They always differ by a multiple of 18. Ormiston triples also exist - see A075093.
"Anagram" means that both primes must not only use the same digits but must use each digit the same number of times. [From Harvey P. Dale, Mar 06 2012]
Dickson's conjecture would imply that the sequence is infinite, e.g. that there are infinitely many k for which 1913+3972900*k and 1931+3972900*k form an Ormiston pair. - Robert Israel, Feb 23 2017
REFERENCES
A. Edwards, Ormiston Pairs, Australian Mathematics Teacher, Vol. 58, No. 2 (2002), pp 12-13.
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Jens Kruse Andersen, Ormiston Tuples
A. Edwards, Ormiston Pairs
Eric Weisstein's World of Mathematics, Rearrangement Prime Pair
EXAMPLE
1913 and 1931 are two successive primes.
Although 179 and 197 are composed of the same digits, they do not form an Ormiston pair as several other primes intervene (i.e. 181, 191, 193).
MAPLE
N:= 10^6: # to get all terms <= N
R:= NULL: p:= 3: q:= 5:
while p <= N do
p:= q;
q:= nextprime(q);
if q-p mod 18 = 0 and sort(convert(p, base, 10)) = sort(convert(q, base, 10)) then
R:= R, p
fi
od:
R; # Robert Israel, Feb 23 2017
MATHEMATICA
Prime[ Select[ Range[10^4], Sort[ IntegerDigits[ Prime[ # ]]] == Sort[ IntegerDigits[ Prime[ # + 1]]] & ]]
a = {1}; b = {2}; Do[b = Sort[ IntegerDigits[ Prime[n]]]; If[a == b, Print[ Prime[n - 1], ", ", Prime[n]]]; a = b, {n, 1, 10^4}]
Transpose[Select[Partition[Prime[Range[8600]], 2, 1], Sort[IntegerDigits[ First[#]]] == Sort[ IntegerDigits[Last[#]]]&]][[1]] (* Harvey P. Dale, Mar 06 2012 *)
PROG
(PARI) is(n)=isprime(n)&&vecsort(Vec(Str(n)))==vecsort(Vec(Str(nextprime(n+1)))) \\ Charles R Greathouse IV, Aug 09 2011
(PARI) p=2; forprime(q=3, 1e5, if((q-p)%18==0&&vecsort(Vec(Str(p)))==vecsort(Vec(Str(q))), print1(p", ")); p=q) \\ Charles R Greathouse IV, Aug 09 2011, minor edits by M. F. Hasler, Oct 11 2012
(Haskell)
import Data.List (sort)
a069567 n = a069567_list !! (n-1)
a069567_list = f a000040_list where
f (p:ps@(p':_)) = if sort (show p) == sort (show p')
then p : f ps else f ps
-- Reinhard Zumkeller, Apr 03 2015
(Python)
from sympy import nextprime
from itertools import islice
def agen(): # generator of terms
p, hp, q, hq = 2, "2", 3, "3"
while True:
if hp == hq: yield p
p, q = q, nextprime(q)
hp, hq = hq, "".join(sorted(str(q)))
print(list(islice(agen(), 38))) # Michael S. Branicky, Feb 19 2024
CROSSREFS
KEYWORD
nonn,base,nice
AUTHOR
Amarnath Murthy, Mar 24 2002
EXTENSIONS
Comments and references from Andy Edwards (AndynGen(AT)aol.com), Jul 09 2002
Edited by Robert G. Wilson v, Jul 15 2002 and Aug 29 2002
Minor edits by Ray Chandler, Jul 16 2009
STATUS
approved