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A072274
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List of Ormiston prime pairs.
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14
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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Andy Edwards, Ormiston Pairs, Australian Mathematics Teacher, Vol. 58, No. 2 (2002), pp. 12-13.
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FORMULA
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EXAMPLE
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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.)
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MATHEMATICA
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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}]
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PROG
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(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
(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)))
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CROSSREFS
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Cf. A075093 (smallest member of Ormiston prime triple), A161160 (smallest member of Ormiston prime quadruple).
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KEYWORD
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base,nonn
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AUTHOR
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Andy Edwards (AndynGen(AT)aol.com), Jul 09 2002
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EXTENSIONS
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STATUS
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approved
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