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%I #8 Jan 16 2014 04:49:23
%S 0,0,0,0,0,2,3,2,4,4,4,4,4,4,2,6,3,7,4,2,7,3,5,4,4,6,6,6,4,4,7,8,9,6,
%T 6,11,8,10,6,6,12,8,13,6,12,8,13,10,7,14,10,11,11,11,16,14,13,9,15,11,
%U 23,14,12,11,12,10,14,8,15,9,14,13,11,12,9,19,9,14,11,16,8,14,5,13,8,13,9,13,10,15,10,11,12,17,8,13,10,11,7,18
%N a(n) = |{0 < k < n - 2: 2*m + 1, m*(m+1) - prime(m) and m*(m+1) - prime(m+1) are all prime with m = phi(k) + phi(n-k)/2}|, where phi(.) is Euler's totient function.
%C Conjecture: a(n) > 0 for all n > 5.
%C This implies that there are infinitely many odd primes p = 2*m + 1 with m*(m+1) - prime(m) and m*(m+1)- prime(m+1) both prime.
%H Zhi-Wei Sun, <a href="/A235805/b235805.txt">Table of n, a(n) for n = 1..10000</a>
%e a(8) = 2 since phi(4) + phi(4)/2 = 3 with 2*3 + 1 = 7, 3*4 - prime(3) = 7 and 3*4 - prime(4) = 5 all prime, and phi(5) + phi(3)/2 = 5 with 2*5 + 1 = 11, 5*6 - prime(5) = 19 and 5*6 - prime(6) = 17 all prime.
%t q[n_]:=PrimeQ[2n+1]&&PrimeQ[n(n+1)-Prime[n]]&&PrimeQ[n(n+1)-Prime[n+1]]
%t f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/2
%t a[n_]:=Sum[If[q[f[n,k]],1,0],{k,1,n-3}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000010, A000040, A235592, A235728, A235806.
%K nonn
%O 1,6
%A _Zhi-Wei Sun_, Jan 16 2014
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