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A233566 a(n) = |{0 < p < n: p and p*phi(n-p) - 1 are both prime}|, where phi(.) is Euler's totient function (A000010).

%I A233566 #8 Dec 13 2013 18:17:53
%S A233566 0,0,0,1,2,2,2,2,2,4,3,3,4,4,3,3,2,2,4,3,3,5,5,4,5,3,2,6,2,4,2,7,7,8,
%T A233566 5,4,8,4,4,8,5,5,8,4,4,5,6,5,5,10,7,8,4,4,5,6,8,7,4,6,6,9,11,7,10,4,6,
%U A233566 7,8,10,4,7,6,5,5,12,8,8,7,11,13,11,12,5,8,7,11,9,5,8,5,6,12,8,8,5,9,5,11,12
%N A233566 a(n) = |{0 < p < n: p and p*phi(n-p) - 1 are both prime}|, where phi(.) is Euler's totient function (A000010).
%C A233566 Conjecture: a(n) > 0 for all n > 3. Also, for any n > 2 there is a prime p < n with p^2*phi(n-p) - 1 prime.
%H A233566 Zhi-Wei Sun, <a href="/A233566/b233566.txt">Table of n, a(n) for n = 1..10000</a>
%e A233566 a(4) = 1 since 3 and 3*phi(4-3) - 1 = 2 are both prime.
%e A233566 a(5) = 2 since 2 and 2*phi(5-2) - 1 = 3 are both prime, and also 3 and 3*phi(5-3) - 1 = = 2 are both prime.
%t A233566 a[n_]:=Sum[If[PrimeQ[Prime[k]*EulerPhi[n-Prime[k]]-1],1,0],{k,1,PrimePi[n-1]}]
%t A233566 Table[a[n],{n,1,100}]
%Y A233566 Cf. A000010, A000040, A233542, A233547, A233549.
%K A233566 nonn
%O A233566 1,5
%A A233566 _Zhi-Wei Sun_, Dec 13 2013