%I
%S 0,0,0,0,0,0,0,0,0,1,1,1,2,0,1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,2,0,3,
%T 0,1,0,2,3,4,3,3,1,2,3,1,6,2,9,2,5,3,4,3,8,1,4,3,9,2,3,5,6,6,7,3,8,7,
%U 6,4,4,5,7,3,6,5,1,4,6,6,2,3,4,5,4,11,4,5,4,7,2,5,5,5,2,6,2,5,5,7
%N a(n) = {0 < k < n: p = phi(k) + phi(nk)/3 + 1, q = prime(p)  p + 1 and r = prime(q)  q + 1 are all prime}, where phi(.) is Euler's totient function.
%C Conjecture: a(n) > 0 for all n > 37.
%C This implies that there are infinitely many primes p with q = prime(p)  p + 1 and r = prime(q)  q + 1 both prime.
%H ZhiWei Sun, <a href="/A235924/b235924.txt">Table of n, a(n) for n = 1..10000</a>
%H Z.W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014
%e a(20) = 1 since phi(6) + phi(14)/3 + 1 = 5, prime(5)  4 = 11  4 = 7 and prime(7)  6 = 17  6 = 11 are all prime.
%e a(77) = 1 since phi(59) + phi(18)/3 + 1 = 61, prime(61)  60 = 283  60 = 223 and prime(223)  222 = 1409  222 = 1187 are all prime.
%e a(1471) = 1 since phi(25) + phi(1446)/3 + 1 = 181, prime(181)  180 = 1087  180 = 907 and prime(907)  906 = 7057  906 = 6151 are all prime.
%t q[n_]:=Prime[n]n+1
%t f[n_,k_]:=EulerPhi[k]+EulerPhi[nk]/3+1
%t p[n_,k_]:=PrimeQ[f[n,k]]&&PrimeQ[q[f[n,k]]]&&PrimeQ[q[q[f[n,k]]]]
%t a[n_]:=Sum[If[p[n,k],1,0],{k,1,n1}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000010, A000040, A234694, A234695.
%K nonn
%O 1,13
%A _ZhiWei Sun_, Jan 17 2014
