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%I #6 Feb 04 2014 02:48:13
%S 0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,1,3,2,3,2,3,3,3,2,2,4,3,3,2,2,4,3,4,3,
%T 4,4,3,3,4,5,4,1,3,3,5,4,4,4,4,5,3,4,2,4,4,4,5,2,4,1,4,4,4,4,1,3,4,4,
%U 5,5
%N Number of ways to write n = k + m (0 < k < m) with k and m terms of A072281.
%C Conjecture: a(n) > 0 for all n > 11.
%C Clearly, this implies the twin prime conjecture.
%H Zhi-Wei Sun, <a href="/A237127/b237127.txt">Table of n, a(n) for n = 1..10000</a>
%e a(13) = 1 since 13 = 5 + 8 with phi(5) - 1 = 3, phi(5) + 1 = 5, phi(8) - 1 = 3 and phi(8) + 1 = 5 all prime.
%e a(60) = 1 since 60 = 18 + 42 with phi(18) - 1 = 5, phi(18) + 1 = 7, phi(42) - 1 = 11 and phi(42) + 1 = 13 all prime.
%e a(84) = 1 since 84 = 7 + 77 with phi(7) - 1 = 5, phi(7) + 1 = 7, phi(77) - 1 = 59 and phi(77) + 1 = 61 all prime.
%t PQ[n_]:=PrimeQ[EulerPhi[n]-1]&&PrimeQ[EulerPhi[n]+1]
%t a[n_]:=Sum[If[PQ[k]&&PQ[n-k],1,0],{k,1,(n-1)/2}]
%t Table[a[n],{n,1,70}]
%Y Cf. A000010, A000040, A001359, A006512, A014574, A072281.
%K nonn
%O 1,15
%A _Zhi-Wei Sun_, Feb 04 2014
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