A117646 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.

3, 5, 7, 47, 53, 59, 151, 157, 163, 167, 173, 179, 199, 211, 223, 251, 257, 263, 257, 263, 269, 367, 373, 379, 557, 563, 569, 587, 593, 599, 601, 607, 613, 647, 653, 659, 727, 733, 739, 941, 947, 953, 971, 977, 983, 1097, 1103, 1109, 1117, 1123, 1129, 1181

Offset

1,1

Comments

A Goedel prime equivalent to if A Implies B and B implies C then A Implies 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.

Not monotone: a(18) = 263 > 257 = a(19). - Charles R Greathouse IV, Dec 17 2016

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Wikipedia, The War of the Worlds (1953 movie)

Wikipedia, The War of the Worlds (novel)

Formula

a(n) = If gap=2*m then { Prime[n],Prime[n+1],Prime[n+2]}.

Mathematica

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]
Select[Partition[Prime[Range[200]], 3, 1], Length[Union[Differences[#]]] == 1&]// Flatten (* Harvey P. Dale, Dec 23 2018 *)

Prog

(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

Crossrefs

Sequence in context: A102742 A363965 A089044 * A064857 A065913 A137999

Adjacent sequences: A117643 A117644 A117645 * A117647 A117648 A117649

Keyword

nonn,uned

Author

Roger L. Bagula, Apr 10 2006

Status

approved