A093483 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.

2, 4, 8, 14, 38, 98, 344, 22268, 79808, 187124, 347978, 2171618, 4219797674, 98059918334, 22518029924768, 54420534706118, 252534792143648

Offset: 1

Author

Amarnath Murthy, Apr 14 2004

Keywords

• hard
• nonn
• nice

Comments

a(i) == 2 mod 6 for i > 2. - Walter Kehowski, Jun 03 2006

a(i) == either 8 or 14 (mod 30) for i > 2. - Robert G. Wilson v, Oct 16 2012

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

No more terms less than 7*10^12. - David Wasserman, Apr 03 2007

Examples

			a(5) = 38 because 38+2+1, 38+4+1, 38+8+1 and 38+14+1 are all prime.
		

Links

• G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70.
• Carlos Rivera, Puzzle 37. Set of even numbers { ai } such that every ai + aj + 1 is prime ( i & j are different ), The Prime Puzzles and Problems Connection.

Crossrefs

Cf. A034881, A117480, A121404, A103828.

Cf. A010051.

Programs

Haskell

a093483 n = a093483_list !! (n-1)
a093483_list = f ([2..7] ++ [8,14..]) [] where
   f (x:xs) ys = if all (== 1) $ map (a010051 . (+ x)) ys
                    then x : f xs ((x+1):ys) else f xs ys
-- Reinhard Zumkeller, Dec 11 2011

Maple

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

Mathematica

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 *)

Extensions

a(7) from Jonathan Vos Post, Mar 22 2006

More terms from Joshua Zucker, Jul 24 2006

Edited and extended to a(14) by David Wasserman, Apr 03 2007

a(15)-a(17) from Don Reble, added by N. J. A. Sloane, Sep 18 2012