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A069567 Smaller of two consecutive primes which are anagrams of each other.

%I A069567 #55 Apr 25 2025 16:00:27
%S A069567 1913,18379,19013,25013,34613,35617,35879,36979,37379,37813,40013,
%T A069567 40213,40639,45613,48091,49279,51613,55313,56179,56713,58613,63079,
%U A069567 63179,64091,65479,66413,74779,75913,76213,76579,76679,85313,88379,90379,90679,93113,94379,96079
%N A069567 Smaller of two consecutive primes which are anagrams of each other.
%C A069567 Smaller members of Ormiston prime pairs.
%C A069567 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.
%C A069567 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.
%C A069567 "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]
%C A069567 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
%D A069567 Andy Edwards, Ormiston Pairs, Australian Mathematics Teacher, Vol. 58, No. 2 (2002), pp. 12-13.
%H A069567 Charles R Greathouse IV, <a href="/A069567/b069567.txt">Table of n, a(n) for n = 1..10000</a>
%H A069567 Jens Kruse Andersen, <a href="http://primerecords.dk/ormiston_tuples.htm">Ormiston Tuples</a>
%H A069567 Andy Edwards, <a href="https://web.archive.org/web/20200401140701/https://www.aamt.edu.au/content/download/742/19588/file/amt-s.pdf">Ormiston Pairs</a> [Archive machine link]; <a href="/A069567/a069567.pdf">local copy</a> [with permission]
%H A069567 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RearrangementPrimePair.html">Rearrangement Prime Pair</a>.
%e A069567 1913 and 1931 are two successive primes.
%e A069567 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).
%p A069567 N:= 10^6: # to get all terms <= N
%p A069567 R:= NULL: p:= 3: q:= 5:
%p A069567 while p <= N do
%p A069567   p:= q;
%p A069567   q:= nextprime(q);
%p A069567   if q-p mod 18 = 0 and sort(convert(p,base,10)) = sort(convert(q,base,10)) then
%p A069567     R:= R, p
%p A069567   fi
%p A069567 od:
%p A069567 R; # _Robert Israel_, Feb 23 2017
%t A069567 Prime[ Select[ Range[10^4], Sort[ IntegerDigits[ Prime[ # ]]] == Sort[ IntegerDigits[ Prime[ # + 1]]] & ]]
%t A069567 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}]
%t A069567 Transpose[Select[Partition[Prime[Range[8600]],2,1],Sort[IntegerDigits[ First[#]]] == Sort[ IntegerDigits[Last[#]]]&]][[1]] (* _Harvey P. Dale_, Mar 06 2012 *)
%o A069567 (PARI) is(n)=isprime(n)&&vecsort(Vec(Str(n)))==vecsort(Vec(Str(nextprime(n+1)))) \\ _Charles R Greathouse IV_, Aug 09 2011
%o A069567 (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
%o A069567 (Haskell)
%o A069567 import Data.List (sort)
%o A069567 a069567 n = a069567_list !! (n-1)
%o A069567 a069567_list = f a000040_list where
%o A069567    f (p:ps@(p':_)) = if sort (show p) == sort (show p')
%o A069567                      then p : f ps else f ps
%o A069567 -- _Reinhard Zumkeller_, Apr 03 2015
%o A069567 (Python)
%o A069567 from sympy import nextprime
%o A069567 from itertools import islice
%o A069567 def agen(): # generator of terms
%o A069567     p, hp, q, hq = 2, "2", 3, "3"
%o A069567     while True:
%o A069567         if hp == hq: yield p
%o A069567         p, q = q, nextprime(q)
%o A069567         hp, hq = hq, "".join(sorted(str(q)))
%o A069567 print(list(islice(agen(), 38))) # _Michael S. Branicky_, Feb 19 2024
%Y A069567 Cf. A072274, A075093, A161160, A066540.
%Y A069567 Cf. A000040, A028906.
%K A069567 nonn,base,nice
%O A069567 1,1
%A A069567 _Amarnath Murthy_, Mar 24 2002
%E A069567 Comments and references from Andy Edwards (AndynGen(AT)aol.com), Jul 09 2002
%E A069567 Edited by _Robert G. Wilson v_, Jul 15 2002 and Aug 29 2002
%E A069567 Minor edits by _Ray Chandler_, Jul 16 2009