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A224887
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Honaker trios: Consecutive prime numbers p < q < r such that p | (qr+1).
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0
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OFFSET
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1,1
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COMMENTS
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The primes p, q, r are listed as a(3n-2), a(3n-1), a(3n) for n = 1, 2, 3, ...
Caldwell & Cheng show that there are only 3 Honaker trios below 2*10^17 and that these are the only Honaker trios if the Cramér-Granville conjecture is satisfied with a constant M < 199262; they also give other sufficient conditions for the number of Honaker trios to be finite.
Strictly speaking, the keywords "fini,full" are thus only conjectured, but given that the Cramér-Granville conjecture is believed to hold with M = 2*exp(-gamma) ~ 1.123, it seems justified to use them.
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REFERENCES
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R. Crandall and C. Pomerance, Prime Numbers - A Computational Perspective, Springer-Verlag, New York, 2001, p. 73.
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LINKS
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MATHEMATICA
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Flatten[Select[Partition[Prime[Range[50]], 3, 1], Divisible[Times@@ Rest[ #]+1, #[[1]]]&]] (* Harvey P. Dale, Feb 03 2015 *)
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PROG
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(PARI) forprime(p=3, default(primelimit), (nextprime(p+1)*p+1)%precprime(p-1)||print1(precprime(p-1)", "p", "nextprime(p+1)", ")) \\ For PARI/GP version >= 2.6, default(primelimit) can be omitted.
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CROSSREFS
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KEYWORD
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nonn,fini,full
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AUTHOR
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STATUS
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approved
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