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

1,2

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

Links

Table of n, a(n) for n=1..18.

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.

Formula

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

Example

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.

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 *)

Crossrefs

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

Sequence in context: A360703 A191007 A036215 * A110740 A348555 A042938

Adjacent sequences: A103825 A103826 A103827 * A103829 A103830 A103831

Keyword

nonn,more

Author

Walter Kehowski, May 29 2006

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

Status

approved