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A236470 a(n) = |{0 < k < n: p = prime(k) + phi(n-k), p + 2 and prime(p) + 2 are all prime}|, where phi(.) is Euler's totient function.

%I A236470 #7 Jan 03 2024 07:42:06
%S A236470 0,1,1,0,1,1,0,1,0,0,1,1,0,2,1,2,1,1,1,1,0,2,2,2,0,1,0,1,1,0,3,0,1,0,
%T A236470 1,3,0,1,1,1,0,1,0,0,1,2,1,0,0,0,0,2,0,1,0,0,0,0,1,1,0,0,2,1,0,1,1,0,
%U A236470 0,0,0,0,0,0,0,0,2,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0
%N A236470 a(n) = |{0 < k < n: p = prime(k) + phi(n-k), p + 2 and prime(p) + 2 are all prime}|, where phi(.) is Euler's totient function.
%C A236470 Conjecture: a(n) > 0 for all n > 948.
%C A236470 We have verified this for n up to 50000.
%C A236470 The conjecture implies that there are infinitely many primes p with p + 2 and prime(p) + 2 both prime. See A236458 for such primes p.
%H A236470 Zhi-Wei Sun, <a href="/A236470/b236470.txt">Table of n, a(n) for n = 1..10000</a>
%e A236470  a(12) = 1 since prime(5) + phi(7) = 11 + 6 = 17, 17 + 2 = 19 and prime(17) + 2 = 59 + 2 = 61 are all prime.
%e A236470 a(97) = 1 since prime(7) + phi(90) = 17 + 24 = 41, 41 + 2 = 43 and prime(41) + 2 = 179 + 2 = 181 are all prime.
%t A236470 p[n_]:=PrimeQ[n]&&PrimeQ[n+2]&&PrimeQ[Prime[n]+2]
%t A236470 f[n_,k_]:=Prime[k]+EulerPhi[n-k]
%t A236470 a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-1}]
%t A236470 Table[a[n],{n,1,100}]
%Y A236470 Cf. A000010, A000040, A001359, A006512, A236097, A236456, A236458, A236468.
%K A236470 nonn
%O A236470 1,14
%A A236470 _Zhi-Wei Sun_, Jan 26 2014