A161160 -id:A161160 - cp's OEIS frontend

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

1913, 18379, 19013, 25013, 34613, 35617, 35879, 36979, 37379, 37813, 40013, 40213, 40639, 45613, 48091, 49279, 51613, 55313, 56179, 56713, 58613, 63079, 63179, 64091, 65479, 66413, 74779, 75913, 76213, 76579, 76679, 85313, 88379, 90379, 90679, 93113, 94379, 96079

Offset: 1

Amarnath Murthy, Mar 24 2002

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

			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).

Andy Edwards, Ormiston Pairs, Australian Mathematics Teacher, Vol. 58, No. 2 (2002), pp. 12-13.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Jens Kruse Andersen, Ormiston Tuples
Andy Edwards, Ormiston Pairs [Archive machine link]; local copy [with permission]
Eric Weisstein's World of Mathematics, Rearrangement Prime Pair.

Cf. A072274, A075093, A161160, A066540.

Cf. A000040, A028906.

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

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

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 *)

is(n)=isprime(n)&&vecsort(Vec(Str(n)))==vecsort(Vec(Str(nextprime(n+1)))) \\ Charles R Greathouse IV, Aug 09 2011

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

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

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

A072274 List of Ormiston prime pairs.

Original entry on oeis.org

1913, 1931, 18379, 18397, 19013, 19031, 25013, 25031, 34613, 34631, 35617, 35671, 35879, 35897, 36979, 36997, 37379, 37397, 37813, 37831, 40013, 40031, 40213, 40231, 40639, 40693, 45613, 45631, 48091, 48109, 49279, 49297, 51613, 51631, 55313, 55331, 56179, 56197

Offset: 1

Andy Edwards (AndynGen(AT)aol.com), Jul 09 2002

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 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 may exist but must be very large.
The smallest Ormiston triple is (11117123, 11117213, 11117321), the smallest Ormiston quadruple is (6607882123, 6607882213, 6607882231, 6607882321); see Andersen link. - Klaus Brockhaus, Jul 22 2009
The current wording of the definition suggests that the second member of Ormiston prime triples (cf. A075093) is repeated. Indeed, such a triple (p,q,r) corresponds to two pairs (a(2k-1)=p,a(2k)=q) and (a(2k+1)=q,a(2k+2)=r). (If they were listed as ...,p,q,r,...,  then the sequence would still contain both pairs as (non-disjoint) subsequences. But if that was the intended meaning, then one would prefer the title "Members of O. prime pairs" (or simply O. primes?). Under this assumption, a(n)=a(n+1) iff a(n-1)=A075093(k) (for some k) is the smallest member of an Ormiston prime triple (a(n-1), a(n)=a(n+1), a(n+2)). In particular this is the case for the first two elements of Ormiston quadruples, cf. A161160. - M. F. Hasler, Oct 11 2012
The term "Ormiston pair" was coined by Andy Edwards in 2002 after Ormiston College in Queensland, Australia. - Amiram Eldar, Nov 25 2020

			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.)

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..7444 from Klaus Brockhaus)
Jens Kruse Andersen, Ormiston Tuples.
Andy Edwards, Ormiston Pairs, Australian Mathematics Teacher, Vol. 58, No. 2 (2002), pp. 12-13.
Eric Weisstein's World of Mathematics, Rearrangement Prime Pair.

Cf. A069567.

Cf. A075093 (smallest member of Ormiston prime triple), A161160 (smallest member of Ormiston prime quadruple).

&cat[ [ p, q ]: p in PrimesUpTo(52000) | (q-p) mod 18 eq 0 and a eq b where a is Sort(Intseq(p)) where b is Sort(Intseq(q)) where q is NextPrime(p) ]; // Klaus Brockhaus, Jul 22 2009

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}]

is(n)=if(!isprime(n), return(0)); my(d=vecsort(digits(n))); vecsort(digits(precprime(n-1)))==d || vecsort(digits(nextprime(n+1)))==d \\ Charles R Greathouse IV, Mar 07 2016

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 from [p, q]
        p, q = q, nextprime(q)
        hp, hq = hq, "".join(sorted(str(q)))
print(list(islice(agen(), 38))) # Michael S. Branicky, Feb 19 2024

a(2k-1)=A069567(k); a(2k)=nextprime(a(2k-1)+1). - M. F. Hasler, Oct 13 2012

Edited and corrected by Robert G. Wilson v, Jul 15 2002

A075093 Smallest member of Ormiston prime triple.

Original entry on oeis.org

11117123, 12980783, 14964017, 32638213, 32964341, 33539783, 35868013, 44058013, 46103237, 48015013, 50324237, 52402783, 58005239, 60601237, 61395239, 74699789, 76012879, 78163123, 80905879, 81966341, 82324237

Offset: 1

Robert G. Wilson v, Aug 31 2002

A subsequence of A069567 which is in turn a subsequence of A072274. More precisely, this A075093 lists exactly the terms which precede repeated elements in A072274, since an Ormiston triple (p,q,r), i.e., three consecutive primes whose decimal representations are anagrams of each other, corresponds to two pairs (p,q) and (q,r), so that p=a(n) implies p=A072274(2k-1)=A069567(k) for some k, and q=A072274(2k)=A072274(2k+1)=A069567(k+1) and r=A072274(2k+2). - M. F. Hasler, Oct 11 2012

			The first Ormiston triples are 11117123, 11117213 and 11117321.

Reinhard Zumkeller, Table of n, a(n) for n = 1..100

Cf. A069567 - smaller member of an Ormiston prime pair.

Cf. A072274, A161160, A066540. - M. F. Hasler, Oct 11 2012

a075093 n = a075093_list !! (n-1)
a075093_list = f a000040_list where
   f (p:ps@(q:r:_)) =
     if sort (show p) == sort (show q) && sort (show q) == sort (show r)
        then p : f ps else f ps
-- Reinhard Zumkeller, Mar 09 2012

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; a = 0; b = 1; Do[c = NextPrim[b]; If[ Sort[ IntegerDigits[a]] == Sort[ IntegerDigits[b]] == Sort[ IntegerDigits[c]], Print[a]]; a = b; b = c, {n, 1, 10^7}]
op3Q[{a_,b_,c_}]:=Sort[IntegerDigits[a]]==Sort[IntegerDigits[b]] == Sort[ IntegerDigits[ c]]; Transpose[Select[Partition[Prime[ Range[ 5000000]],3,1], op3Q]][[1]] (* Harvey P. Dale, Jun 16 2014 *)

is_A075093(n)={isprime(n) & vecsort(Vec(Str(n)))==vecsort(Vec(Str(n=nextprime(n+1)))) & vecsort(Vec(Str(n)))==vecsort(Vec(Str(nextprime(n+1))))} \\ M. F. Hasler, Oct 11 2012

from sympy import nextprime
from itertools import islice
def hash(n): return "".join(sorted(str(n)))
def agen(start=2): # generator of terms
    p = nextprime(start-1); q=nextprime(p); r=nextprime(q)
    hp, hq, hr = list(map(hash, [p, q, r]))
    while True:
        if hp == hq == hr: yield p
        p, q, r = q, r, nextprime(r)
        hp, hq, hr = hq, hr, hash(r)
print(list(islice(agen(), 3))) # Michael S. Branicky, Feb 19 2024

A217372 Initial prime in the first Ormiston n-tuple.

Original entry on oeis.org

2, 1913, 11117123, 6607882123, 20847942560791

Offset: 1

Jens Kruse Andersen, Oct 20 2012

An Ormiston n-tuple is n consecutive primes containing the same decimal digits in different order. a(5) found by Giovanni Resta. a(6) may be 166389896360719.

			(1913, 1931) is the first case of two consecutive primes with the same digits. The first 3-, 4- and 5-tuples are: (11117123, 11117213, 11117321), (6607882123, 6607882213, 6607882231, 6607882321), (20847942560791, 20847942560917, 20847942560971, 20847942561079, 20847942561097).

Jens Kruse Andersen, Ormiston Tuples
A. Edwards, Ormiston Pairs, Australian Mathematics Teacher, Vol. 58, No. 2 (2002), pp 12-13.
E. W. Weisstein, Rearrangement Prime Pair at MathWorld.Wolfram.com.

Cf. A069567 (Ormiston pairs), A075093 (triples), A161160 (quadruples), A217797 (5-tuples)

A217797 Smallest member of Ormiston prime 5-tuple.

Original entry on oeis.org

20847942560791, 21815124622913, 35581541330719, 40546521517819, 47950363950791, 54808830290791, 65923105730719, 84573572180719, 85950417240719

Offset: 1

Giovanni Resta, Oct 12 2012

Searched up to 10^14.

On 11 October 2012 Jens Kruse Andersen found a 6-tuple starting at 166389896360719, which is likely to be the smallest.

			a(1) is in the sequence since (20847942560791, 20847942560917, 20847942560971, 20847942561079, 20847942561097) are 5 consecutive primes whose decimal representations contain exactly the same digits.

Jens Kruse Andersen, Ormiston Tuples
E. W. Weisstein, MathWorld: Rearrangement Prime Pair

Cf. A072274 (Ormiston pairs), A075093 (Ormiston triples), A161160 (Ormiston quadruples).

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

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A072274 List of Ormiston prime pairs.

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A075093 Smallest member of Ormiston prime triple.

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A217372 Initial prime in the first Ormiston n-tuple.

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A217797 Smallest member of Ormiston prime 5-tuple.

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