|
%I #41 Mar 07 2024 09:13:27
%S 1,3,9,27,69,429,1059,56499,166839,5020059,7681809,274343589,
%T 8316187179,2866819175649,7180244842749,216549352241349,
%U 22129340663539629,2504509324460255499
%N 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.
%C Is the sequence infinite? Is each prime a(i)+a(j)+1, i<>j, always distinct?
%C Except for a(1), a(n) == 3 (mod 6). - _Robert G. Wilson v_, Jun 02 2006.
%C 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
%C 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
%H G. H. Hardy and J. E. Littlewood, <a href="https://doi.org/10.1007/BF02403921">Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes</a>, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70.
%F a(n) = (A115760(n) - 1)/2.
%e 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.
%p 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;
%t 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 *)
%Y Cf. A093483, A115760, A115782 (primes arising from this sequence), A118818, A128933 (a(n)+1), A291163.
%K nonn,more
%O 1,2
%A _Walter Kehowski_, May 29 2006
%E a(12) from _Robert G. Wilson v_, Jun 03 2006
%E a(13) from _Walter Kehowski_, Jun 03 2006
%E Definition corrected by _Walter Kehowski_, Nov 03 2008
%E a(14)-a(16) from _Don Reble_ added by _N. J. A. Sloane_, Sep 18 2012
%E a(17)-a(18) from _Don Reble_, Aug 17 2021
|
