A236468 - OEIS

%I #5 Jan 26 2014 12:22:34

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

%T 5,3,4,0,6,6,1,3,1,5,4,5,2,5,1,7,1,3,2,5,1,4,1,7,0,5,4,1,8,1,5,5,1,2,

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

%N Number of ordered ways to write n = k + m with k > 0 and m > 0 such that p = phi(k) + phi(m)/2 - 1, p + 2 and prime(p) - 2 are all prime, where phi(.) is Euler's totient function.

%C Conjecture: a(n) > 0 for every n = 250, 251, ....

%C This implies that there are infinitely many twin prime pairs {p, p + 2} with {prime(p) - 2, prime(p)} also a twin prime pair. It is stronger than the twin prime conjecture.

%H Zhi-Wei Sun, <a href="/A236468/b236468.txt">Table of n, a(n) for n = 1..10000</a>

%e  a(33) = 1 since 33 = 7 + 26 with phi(7) + phi(26)/2 - 1 = 11, 11 + 2 = 13 and prime(11) - 2 = 31 - 2 = 29 all prime.

%e a(278) = 1 since 278 = 61 + 217 with phi(61) + phi(217)/2 - 1 = 60 + 90 - 1 = 149, 149 + 2 = 151 and prime(149) - 2 = 859 - 2 = 857 all prime.

%t p[n_]:=PrimeQ[n]&&PrimeQ[n+2]&&PrimeQ[Prime[n]-2]

%t f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/2-1

%t a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-3}]

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

%Y Cf. A000010, A000040, A001359, A006512, A236097, A236456, A236460, A236467.

%K nonn

%O 1,8

%A _Zhi-Wei Sun_, Jan 26 2014