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%I #8 Jan 19 2019 04:14:59
%S 0,0,0,0,0,0,0,1,1,0,2,1,1,2,1,3,2,2,0,2,3,1,2,1,3,3,2,2,1,1,1,3,0,2,
%T 3,2,1,3,0,2,0,1,1,1,1,2,0,0,0,0,2,2,0,1,0,0,0,0,0,0,0,0,1,1,0,1,1,2,
%U 1,0,0,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1
%N a(n) = |{0 < k < n-2: p = 2*phi(k) + phi(n-k)/2 - 1, p + 2, p + 6 and prime(p) + 6 are all prime}|, where phi(.) is Euler's totient function.
%C Conjecture: a(n) > 0 for all n > 146.
%C We have verified this for n up to 52000.
%C The conjecture implies that there are infinitely many prime triples {p, p + 2, p + 6} with {prime(p), prime(p) + 6} a sexy prime pair. See A236509 for such primes p.
%H Zhi-Wei Sun, <a href="/A236508/b236508.txt">Table of n, a(n) for n = 1..10000</a>
%e a(13) = 1 since 2*phi(3) + phi(10)/2 - 1 = 5, 5 + 2 = 7, 5 + 6 = 11 and prime(5) + 6 = 11 + 6 = 17 are all prime.
%e a(244) = 1 since 2*phi(153) + phi(244-153)/2 - 1 = 2*96 + 72/2 - 1 = 227, 227 + 2 = 229, 227 + 6 = 233 and prime(227) + 6 = 1433 + 6 = 1439 are all prime.
%t p[n_]:=PrimeQ[n]&&PrimeQ[n+2]&&PrimeQ[n+6]&&PrimeQ[Prime[n]+6]
%t f[n_,k_]:=2*EulerPhi[k]+EulerPhi[n-k]/2-1
%t a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-3}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000010, A000040, A022004, A023201, A046117, A236464, A236472, A236509.
%K nonn
%O 1,11
%A _Zhi-Wei Sun_, Jan 27 2014
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