A235187 - OEIS

%I #6 Jan 04 2014 12:56:39

%S 0,0,1,1,2,1,1,2,2,3,2,3,2,1,2,2,2,3,1,2,5,5,2,5,3,2,5,2,1,6,2,4,4,1,

%T 5,3,4,3,6,6,3,5,5,2,9,3,3,7,2,4,7,6,3,6,7,5,4,4,4,12,3,2,5,3,3,9,3,1,

%U 7,4,2,8,6,3,8,3,4,7,6,3,10,3,3,10,8,3,11,5,3,10,6,1,9,8,2,7,4,3,9,4

%N Number of ordered ways to write 2*n = p + q with p, q and prime(p) + q - 1 all prime.

%C Conjecture: (i) a(n) > 0 for all n > 2.

%C (ii) Any integer n > 4 can be written as p + q with q > 0 such that p and p - 1 + prime(q) are both prime.

%C (iii) Each integer n > 7 can be written as p + q with q > 0 such that prime(p) + sigma(q) is prime, where sigma(q) denotes the sum of all positive divisors of q.

%C Clearly, part (i) is stronger than Goldbach's conjecture.

%H Zhi-Wei Sun, <a href="/A235187/b235187.txt">Table of n, a(n) for n = 1..10000</a>

%e  a(7) = 1 since 2*7 = 7 + 7 with 7 and prime(7) + 7 - 1 = 17 + 6 = 23 both prime.

%e a(14) = 1 since 2*14 = 11 + 17 with 11, 17 and prime(11) + 16 = 47 all prime.

%e a(92) = 1 since 2*92 = 47 + 137 with 47, 137 and prime(47) + 136 = 347 all prime.

%t a[n_]:=Sum[If[PrimeQ[2n-Prime[k]]&&PrimeQ[Prime[Prime[k]]+2n-Prime[k]-1],1,0],{k,1,PrimePi[2n-1]}]

%t Table[a[n],{n,1,100}]

%Y Cf. A000040, A000203, A045917, A234694.

%K nonn

%O 1,5

%A _Zhi-Wei Sun_, Jan 04 2014