A224887 Honaker trios: Consecutive prime numbers p < q < r such that p | (qr+1).

2, 3, 5, 3, 5, 7, 61, 67, 71

Offset

1,1

Comments

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.

References

R. Crandall and C. Pomerance, Prime Numbers - A Computational Perspective, Springer-Verlag, New York, 2001, p. 73.

Links

Table of n, a(n) for n=1..9.

Chris K. Caldwell, Yuanyou Cheng, Determining Mills' Constant and a Note on Honaker's Problem, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.1

Mathematica

Flatten[Select[Partition[Prime[Range[50]], 3, 1], Divisible[Times@@ Rest[ #]+1, #[[1]]]&]] (* Harvey P. Dale, Feb 03 2015 *)

Prog

(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.

Crossrefs

Sequence in context: A083776 A122820 A261324 * A151571 A193957 A336746

Adjacent sequences: A224884 A224885 A224886 * A224888 A224889 A224890

Keyword

nonn,fini,full

Author

M. F. Hasler, Jul 24 2013

Status

approved