A103828 Sequence of odd numbers defined recursively by: a(1)=1 and a(n) is the first odd number greater than a(n-1) such that a(n) + a(i) + 1 is prime for 1<=i<=n-1.

1, 3, 9, 27, 69, 429, 1059, 56499, 166839, 5020059, 7681809, 274343589, 8316187179, 2866819175649, 7180244842749, 216549352241349, 22129340663539629, 2504509324460255499

Offset: 1

Author

Walter Kehowski, May 29 2006

Keywords

• nonn
• more

Comments

Is the sequence infinite? Is each prime a(i)+a(j)+1, i<>j, always distinct?

Except for a(1), a(n) == 3 (mod 6). - Robert G. Wilson v, Jun 02 2006.

The Hardy-Littlewood k-tuple conjecture would imply that this sequence is infinite. Note that, for n>2, a(n)+2 and a(n)+4 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

From the mod 30 property of A115760 we conclude that a(n) == 9 (mod 15) for n>4. This implies that either a(n) == 9 (mod 30) or == 24 (mod 30), but == 24 (mod 30) is impossible because then == 0 (mod 6). Therefore a(n) == 9 (mod 30) for n>4. - Don Reble, Aug 17 2021

Examples

			a(1)=1, a(2)=3, but 5+1+1=7, 5+3+1=9; 7+1+1=9, 7+3+1=11; 9+1+1=11, 9+3+1=13 so a(3)=9.
		

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.

Crossrefs

Cf. A093483, A115760, A115782 (primes arising from this sequence), A118818, A128933 (a(n)+1), A291163.

Programs

Maple

EP:=[]: for w to 1 do for n from 1 to 8*10^6 do s:=2*n-1; Q:=map(z->z+s+1, EP); if andmap(isprime,Q) then EP:=[op(EP),s]; print(nops(EP),s); fi od od; EP;

Mathematica

a[1] = 1; a[2] = 3; a[n_] := a[n] = Block[{k = a[n - 1] + 6, t = Table[ a[i], {i, n - 1}] + 1}, While[ First@ Union@ PrimeQ[k + t] == False, k += 6]; k]; Do[ Print[ a[n]], {n, 15}] (* Robert G. Wilson v, Jun 03 2006 *)

Formula

a(n) = (A115760(n) - 1)/2.

Extensions

a(12) from Robert G. Wilson v, Jun 03 2006

a(13) from Walter Kehowski, Jun 03 2006

Definition corrected by Walter Kehowski, Nov 03 2008

a(14)-a(16) from Don Reble added by N. J. A. Sloane, Sep 18 2012

a(17)-a(18) from Don Reble, Aug 17 2021