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a(n) = |{0 < k < n: p = prime(k) + phi(n-k) is a Sophie prime having 2 as a primitive root}|.
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%I #8 Jan 18 2014 13:23:44

%S 0,1,1,0,1,1,0,1,1,1,0,2,0,1,0,2,2,2,1,1,1,2,2,2,1,2,3,1,2,2,0,4,3,3,

%T 1,3,0,2,2,2,4,1,4,2,1,4,3,4,6,1,2,5,2,4,5,4,4,1,5,4,1,4,1,3,2,3,4,2,

%U 3,5,2,4,5,2,6,2,2,5,1,4,8,4,3,3,2,3,4,5,5,3,7,2,4,3,5,6,7,6,3,4

%N a(n) = |{0 < k < n: p = prime(k) + phi(n-k) is a Sophie prime having 2 as a primitive root}|.

%C Conjecture: (i) a(n) > 0 for all n > 37.

%C (ii) For any integer n > 7, there is a positive integer k < n such that p = phi(k) + phi(n-k)/2 - 1 is a prime having 2 as a primitive root.

%C Part (i) of the conjecture implies that there are infinitely many Sophie Germain primes p having 2 as a primitive root modulo p.

%C We also have some other similar conjectures.

%H Zhi-Wei Sun, <a href="/A235987/b235987.txt">Table of n, a(n) for n = 1..10000</a>

%e a(10) = 1 since prime(3) + phi(7) = 5 + 6 = 11 is a Sophie prime and 2 is a primitive root modulo 11.

%e a(79) = 1 since prime(19) + phi(60) = 67 + 16 = 83 is a Sophie Germain prime and 2 is a primitive root modulo 83.

%t p[n_]:=PrimeQ[n]&&PrimeQ[2n+1]&&PrimitiveRoot[n]==2

%t f[n_,k_]:=Prime[k]+EulerPhi[n-k]

%t a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-1}]

%t Table[a[n],{n,1,100}]

%Y Cf. A000010, A000040, A001122, A005384.

%K nonn

%O 1,12

%A _Zhi-Wei Sun_, Jan 17 2014