

A093483


a(1) = 2; for n>1, a(n) = smallest integer > a(n1) such that a(n) + a(i) + 1 is prime for all 1 <= i <= n1.


9



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


REFERENCES

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. 44 (1923), 170.


LINKS

Table of n, a(n) for n=1..17.
Prime Puzzles and Problems, Set of even numbers { ai } such that every ai + aj + 1 is prime ( i & j are different ).


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]; k1]; Array[f, 13] (* Robert G. Wilson v, Oct 16 2012 *)


PROG

(Haskell)
a093483 n = a093483_list !! (n1)
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: A337500 A061297 A130711 * A028398 A155249 A118884
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



