A234809 - OEIS

%I #9 Dec 31 2013 04:01:14

%S 0,0,1,2,1,3,1,4,1,1,1,5,3,7,3,1,1,7,5,9,4,2,1,9,5,2,4,3,1,10,5,14,2,

%T 2,2,1,6,14,5,4,1,15,5,16,5,5,3,17,8,4,5,6,3,17,7,5,2,6,6,17,11,25,3,

%U 5,3,1,11,25,4,4,4,22,10,26,6,7,8,3,9,26,7,9,6,25,8,3,7,9,10,25,15,6,2,9,9,2,13,29,3,7

%N a(n) = |{0 < k < n: p = k + phi(n-k) and 2*(n-p) + 1 are both prime}|, where phi(.) is Euler's totient function.

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

%C Clearly, this implies Lemoine's conjecture which states that any odd number 2*n + 1 > 5 can be written as 2*p + q with p and q both prime.

%C See also A234808 for a similar conjecture.

%H Zhi-Wei Sun, <a href="/A234809/b234809.txt">Table of n, a(n) for n = 1..10000</a>

%e a(5) = 1 since 1 + phi(4) = 3 and 2*(5-3) + 1 = 5 are both prime.

%e a(16) = 1 since 7 + phi(9) = 13 and 2*(16-13) + 1 = 7 are both prime.

%e a(41) = 1 since 7 +phi(34) = 23 and 2*(41-23) + 1 = 37 are both prime.

%e a(156) = 1 since 131 + phi(25) = 151 and 2*(156-151) + 1 = 11 are both prime.

%t f[n_,k_]:=k+EulerPhi[n-k]

%t p[n_,k_]:=PrimeQ[f[n,k]]&&PrimeQ[2*(n-f[n,k])+1]

%t a[n_]:=a[n]=Sum[If[p[n,k],1,0],{k,1,n-1}]

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

%Y Cf. A000010, A000040, A046927, A234470, A234475, A234514, A234567, A234615, A234694, A234808

%K nonn

%O 1,4

%A _Zhi-Wei Sun_, Dec 30 2013