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a(n) = |{0 < k < n: m*prime(m) - 1 and m*prime(m) + 1 are both prime with m = sigma(k) + phi(n-k)}|, where sigma(k) is the sum of all positive divisors of k and phi(.) is Euler's totient function.
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%I #19 Jan 14 2014 11:48:40

%S 0,1,1,0,0,0,0,0,0,0,0,0,0,0,1,1,0,1,0,0,1,1,2,1,1,0,0,1,1,6,1,2,2,0,

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

%U 0,1,1,14,3,2,2,2,2,3,4,5,3,2,3,1,3,3,4,6,3,0,5,3,1,0,5,2,0,3,6,1

%N a(n) = |{0 < k < n: m*prime(m) - 1 and m*prime(m) + 1 are both prime with m = sigma(k) + phi(n-k)}|, where sigma(k) is the sum of all positive divisors of k and phi(.) is Euler's totient function.

%C Conjecture: a(n) > 0 for all n > 214.

%C This implies that there are infinitely many twin prime pairs of the special form {m*prime(m) - 1, m*prime(m) + 1}.

%C We have verified the conjecture for n up to 10^5.

%H Zhi-Wei Sun, <a href="/A232442/b232442.txt">Table of n, a(n) for n = 1..10000</a>

%e a(25) = 1 since sigma(6) + phi(19) = 12 + 18 = 30 with {30*prime(30) - 1, 30*prime(30) + 1} = {3389, 3391} a twin prime pair.

%e a(100) = 1 since sigma(75) + phi(25) = 124 + 20 = 144 with {144*prime(144) - 1, 144*prime(144) + 1} = {119087, 119089} a twin prime pair.

%t sigma[n_]:=DivisorSigma[1,n]

%t q[n_]:=PrimeQ[n*Prime[n]-1]&&PrimeQ[n*Prime[n]+1]

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

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

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

%Y Cf. A000010, A000040, A000203, A085637, A086172, A086173, A232270, A232861, A233544, A233539.

%K nonn

%O 1,23

%A _Zhi-Wei Sun_, Jan 14 2014