A072274 List of Ormiston prime pairs.

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

Author

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

Keywords

• base
• nonn

Comments

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

Examples

			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.)
		

Links

• 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.

Crossrefs

Cf. A069567.

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

Programs

Magma

&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

Mathematica

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}]

PARI

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

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 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

Formula

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

Extensions

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