%I #14 Aug 09 2021 21:10:33
%S 6,13,14,15,74,190,6456,40082,251735,251736,251738,637325,637326,
%T 637342,637343,4124456,4124458,10553425,10553433,10553818,27067038,
%U 27067053,69709872,69709877,69709945,69709954,179992917,179992920,179993011,179993025,179993170
%N Numbers k that divide the average of prime(k-1) and prime(k).
%C Each cluster of candidates is about e (2.71828...) times as large as the previous one.
%e The average of prime(6-1) = 11 and prime(6) = 13 is 12, which is divisible by 6, so 6 is a term of the sequence.
%t a = 1; b = 2; Do[ b = Prime[ n ]; If[ IntegerQ[ (a + b)/(2n) ], Print[ n ] ]; a = b, {n, 1, 2*10^8} ]
%t With[{nn=30000000},Last[#]&/@Select[Thread[{Partition[Prime[Range[nn]],2,1],Range[2,nn]}], Divisible[ Mean[First[#]],Last[#]]&]] (* _Harvey P. Dale_, Oct 21 2011 *) (* The program takes a long time to generate the first 22 terms of the sequence; to generate them all, by increasing nn, may overwhelm computer resources and operator patience *)
%Y Cf. A066706.
%K nonn
%O 1,1
%A _Robert G. Wilson v_, Jan 20 2002
%E a(29)-a(31) from _D. S. McNeil_, Oct 22 2011
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