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

1,1

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

Links

Table of n, a(n) for n=1..17.

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.

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A034881, A117480, A121404, A103828.

Cf. A010051.

Sequence in context: A061297 A130711 A355189 * A028398 A359695 A155249

Adjacent sequences: A093480 A093481 A093482 * A093484 A093485 A093486

Keyword

hard,nonn,nice

Author

Amarnath Murthy, Apr 14 2004

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

Status

approved