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%I #20 Jan 19 2019 04:14:58
%S 3,5,7,47,53,59,151,157,163,167,173,179,199,211,223,251,257,263,257,
%T 263,269,367,373,379,557,563,569,587,593,599,601,607,613,647,653,659,
%U 727,733,739,941,947,953,971,977,983,1097,1103,1109,1117,1123,1129,1181
%N Sets of three consecutive primes with equal gaps: prime(n) + 2*m = prime(n+1) and prime(n+1) + 2*m = prime(n+2) for some m.
%C A Goedel prime equivalent to if A Implies B and B implies C then A Implies C.
%C In H. G. Wells's War of the Worlds, the Martians use a base-three number system: in such a system 3^n+2 instead of 2^n+1 primes would be important. Likewise instead of pairs of primes, triples of primes would be studied as "interesting", so I call these Martian Prime triples as that's what gave me the idea for finding them.
%C Not monotone: a(18) = 263 > 257 = a(19). - _Charles R Greathouse IV_, Dec 17 2016
%H Harvey P. Dale, <a href="/A117646/b117646.txt">Table of n, a(n) for n = 1..1000</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/The_War_of_the_Worlds_(1953_movie)">The War of the Worlds (1953 movie)</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/The_War_of_the_Worlds_(novel)">The War of the Worlds (novel)</a>
%F a(n) = If gap=2*m then { Prime[n],Prime[n+1],Prime[n+2]}.
%t a = Delete[Union[Flatten[Table[If[(Prime[n] + 2*m - Prime[n + 1] == 0) && (Prime[n + 1] + 2*m - Prime[ n + 2] == 0), {Prime[n], Prime[n + 1], Prime[ n + 2]}, {}], {m, 1, 17}, {n, 1, 200}], 1]], 1] Flatten[a]
%t Select[Partition[Prime[Range[200]],3,1],Length[Union[Differences[#]]] == 1&]// Flatten (* _Harvey P. Dale_, Dec 23 2018 *)
%o (PARI) p=2;q=3; forprime(r=5,1e4, if(q-p==r-q, print1(p", "q", "r", ")); p=q; q=r) \\ _Charles R Greathouse IV_, Dec 17 2016
%K nonn,uned
%O 1,1
%A _Roger L. Bagula_, Apr 10 2006
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