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%I #19 Feb 03 2015 12:55:23
%S 2,3,5,3,5,7,61,67,71
%N Honaker trios: Consecutive prime numbers p < q < r such that p | (qr+1).
%C The primes p, q, r are listed as a(3n-2), a(3n-1), a(3n) for n = 1, 2, 3, ...
%C 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.
%C 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.
%D R. Crandall and C. Pomerance, Prime Numbers - A Computational Perspective, Springer-Verlag, New York, 2001, p. 73.
%H Chris K. Caldwell, Yuanyou Cheng, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Caldwell/caldwell78.html">Determining Mills' Constant and a Note on Honaker's Problem</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.1
%t Flatten[Select[Partition[Prime[Range[50]],3,1],Divisible[Times@@ Rest[ #]+1, #[[1]]]&]] (* _Harvey P. Dale_, Feb 03 2015 *)
%o (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.
%K nonn,fini,full
%O 1,1
%A _M. F. Hasler_, Jul 24 2013
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