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A233918 a(n) = |{0 < k <= n/2: (phi(k) + phi(n-k))/2 is prime}|, where phi(.) is Euler's totient function.

%I A233918 #13 Dec 21 2013 09:47:14
%S A233918 0,0,0,0,0,1,1,2,2,1,2,3,1,3,2,4,3,2,7,1,3,3,4,7,2,4,5,5,5,5,6,6,4,7,
%T A233918 5,6,4,4,11,5,5,5,11,4,3,5,7,12,4,6,11,3,6,7,8,6,7,8,11,10,5,9,7,9,5,
%U A233918 4,14,8,9,6,10,7,6,10,9,10,7,10,11,7,7,13,11,13,5,8,11,9,9,3,12,4,11,13,11,19,8,12,11,7
%N A233918 a(n) = |{0 < k <= n/2: (phi(k) + phi(n-k))/2 is prime}|, where phi(.) is Euler's totient function.
%C A233918 Conjecture: (i) a(n) > 0 for all n > 5.
%C A233918 (ii) If n > 5 is not equal to 19, then phi(k) + phi(n-k) - 1 and phi(k) + phi(n-k) + 1 are both prime for some 0 < k < n.
%C A233918 (iii) If n > 5, then (phi(k)/2)^2 + (phi(n-k)/2)^2 is prime for some 0 < k < n.
%C A233918 (iv) If n > 8, then (sigma(k) + phi(n-k))/2 is prime for some 0 < k < n, where sigma(k) is the sum of all positive divisors of k.
%H A233918 Zhi-Wei Sun, <a href="/A233918/b233918.txt">Table of n, a(n) for n = 1..10000</a>
%H A233918 Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;29f53124.1312">New representation problems involving Euler's totient function</a>, a message to Number Theory List, Dec. 18, 2013.
%e A233918 a(6) = 1 since (phi(3) + phi(3))/2 = 2 is prime.
%e A233918 a(7) = 1 since (phi(3) + phi(4))/2 = 2 is prime.
%e A233918 a(10) = 1 since (phi(4) + phi (6))/2 = 2 is prime.
%e A233918 a(13) = 1 since (phi(3) + phi(10))/2 = 3 is prime.
%e A233918 a(20) = 1 since (phi(4) + phi(16))/2 = 5 is prime.
%t A233918 a[n_]:=Sum[If[PrimeQ[(EulerPhi[k]+EulerPhi[n-k])/2],1,0],{k,1,n/2}]
%t A233918 Table[a[n],{n,1,100}]
%Y A233918 Cf. A000010, A000040, A233542, A233544, A233547, A233566, A233567, A233867, A234200.
%K A233918 nonn
%O A233918 1,8
%A A233918 _Zhi-Wei Sun_, Dec 21 2013