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%I #8 Dec 30 2013 22:50:26
%S 0,1,1,2,2,3,2,0,3,1,2,5,2,1,5,1,2,7,2,1,4,1,2,1,4,1,4,2,4,11,4,2,3,1,
%T 5,2,3,2,6,1,5,15,4,2,9,1,6,2,5,4,6,4,4,3,8,3,6,4,7,21,2,4,7,1,7,4,6,
%U 4,6,4,8,22,7,3,13,1,10,5,3,5,7,4,9,5,10,5,8,7,7,6,8,5,6,3,8,6,7,4,8,4
%N a(n) = |{0 < k < n: p = k + phi(n-k) and 2*n - p are both prime}|, where phi(.) is Euler's totient function.
%C Conjecture: a(n) > 0 except for n = 1, 8.
%C Clearly, this implies Goldbach's conjecture.
%H Zhi-Wei Sun, <a href="/A234808/b234808.txt">Table of n, a(n) for n = 1..10000</a>
%e a(3) = 1 since 2 + phi(1) = 3 and 2*3 - 3 = 3 are both prime.
%e a(20) = 1 since 11 + phi(9) = 17 and 2*20 - 17 = 23 are both prime.
%e a(22) = 1 since 1 + phi(21) = 13 and 2*22 - 13 = 31 are both prime.
%e a(24) = 1 since 9 + phi(15) = 17 and 2*24 - 17 = 31 are both prime.
%e a(76) = 1 since 67 + phi(9) = 73 and 2*76 - 73 = 79 are both prime.
%t f[n_,k_]:=k+EulerPhi[n-k]
%t p[n_,k_]:=PrimeQ[f[n,k]]&&PrimeQ[2n-f[n,k]]
%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, A002372, A002375, A233547, A233918, A234200, A234246, A234360, A234470, A234475, A234514, A234567, A234615, A234694
%K nonn
%O 1,4
%A _Zhi-Wei Sun_, Dec 30 2013
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