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%I #10 Jan 13 2014 16:52:20
%S 0,0,0,1,2,1,1,3,2,1,3,1,1,2,2,3,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,3,0,1,2,2,
%U 1,1,1,2,1,2,1,1,1,2,1,4,6,3,6,0,6,4,5,3,1,3,4,2,3,4,1,8,6,4,8,8
%N Number of ways to write n = k + m with k > 0 and m > 2 such that p = phi(k) + phi(m)/2 + 1, prime(p) - p + 1 and p*(p+1) - prime(p) are all prime, where phi(.) is Euler's totient function.
%C Conjecture: a(n) > 0 for all n > 84.
%C Clearly, this implies that there are infinitely many primes p with prime(p) - p + 1 and p*(p+1) - prime(p) both prime.
%H Zhi-Wei Sun, <a href="/A235682/b235682.txt">Table of n, a(n) for n = 1..10000</a>
%e a(10) = 1 since 10 = 1 + 9 with phi(1) + phi(9)/2 + 1 = 5, prime(5) - 5 + 1 = 7 and 5*6 - prime(5) = 19 all prime.
%e a(95) = 1 since 95 = 62 + 33 with phi(62) + phi(33)/2 + 1 = 41, prime(41) - 41 + 1 = 139 and 41*42 - prime(41) = 1543 all prime.
%e a(421) = 1 since 421 = 289 + 132 with phi(289) + phi(132)/2 + 1 = 293, prime(293) - 293 + 1 = 1621 and 293*294 - prime(293) = 84229 all prime.
%t PQ[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]-n+1]&&PrimeQ[n(n+1)-Prime[n]]
%t f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/2+1
%t a[n_]:=Sum[If[PQ[f[n,k]],1,0],{k,1,n-3}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000010, A000040, A232353, A234694, A234695, A235189, A235330, A235613, A235614, A235661, A235681.
%K nonn
%O 1,5
%A _Zhi-Wei Sun_, Jan 13 2014
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