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%I #37 Jul 23 2021 10:14:04
%S 2,4,8,14,38,98,344,22268,79808,187124,347978,2171618,4219797674,
%T 98059918334,22518029924768,54420534706118,252534792143648
%N a(1) = 2; for n>1, a(n) = smallest integer > a(n-1) such that a(n) + a(i) + 1 is prime for all 1 <= i <= n-1.
%C a(i) == 2 mod 6 for i > 2. - _Walter Kehowski_, Jun 03 2006
%C a(i) == either 8 or 14 (mod 30) for i > 2. - _Robert G. Wilson v_, Oct 16 2012
%C The Hardy-Littlewood k-tuple conjecture would imply that this sequence is infinite. Note that, for n > 2, a(n)+3 and a(n)+5 are both primes, so a proof that this sequence is infinite would also show that there are infinitely many twin primes. - _N. J. A. Sloane_, Apr 21 2007
%C No more terms less than 7*10^12. - _David Wasserman_, Apr 03 2007
%H G. H. Hardy and J. E. Littlewood, <a href="https://doi.org/10.1007/BF02403921">Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes</a>, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70.
%H Carlos Rivera, <a href="http://primepuzzles.net/puzzles/puzz_037.htm">Puzzle 37. Set of even numbers { ai } such that every ai + aj + 1 is prime ( i & j are different )</a>, The Prime Puzzles and Problems Connection.
%e a(5) = 38 because 38+2+1, 38+4+1, 38+8+1 and 38+14+1 are all prime.
%p EP:=[2,4]: P:=[]: for w to 1 do for n from 1 to 800*10^6 do s:=6*n+2; Q:=map(z-> z+s+1); if andmap(isprime,Q) then EP:=[op(EP),s]; P:=[op(P),op(Q)] fi; od od; EP; P: # _Walter Kehowski_, Jun 03 2006
%t f[1] = 2; f[2] = 4; f[3] = 8; f[n_] := f[n] = Block[{lst = Array[f, n - 1], k = f[n - 1] + 7}, While[ Union[ PrimeQ[k + lst]] != {True}, k += 6]; k-1]; Array[f, 13] (* _Robert G. Wilson v_, Oct 16 2012 *)
%o (Haskell)
%o a093483 n = a093483_list !! (n-1)
%o a093483_list = f ([2..7] ++ [8,14..]) [] where
%o f (x:xs) ys = if all (== 1) $ map (a010051 . (+ x)) ys
%o then x : f xs ((x+1):ys) else f xs ys
%o -- _Reinhard Zumkeller_, Dec 11 2011
%Y Cf. A034881, A117480, A121404, A103828.
%Y Cf. A010051.
%K hard,nonn,nice
%O 1,1
%A _Amarnath Murthy_, Apr 14 2004
%E a(7) from _Jonathan Vos Post_, Mar 22 2006
%E More terms from _Joshua Zucker_, Jul 24 2006
%E Edited and extended to a(14) by _David Wasserman_, Apr 03 2007
%E a(15)-a(17) from _Don Reble_, added by _N. J. A. Sloane_, Sep 18 2012
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