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%I #18 Apr 16 2018 03:02:22
%S 0,0,0,0,1,2,3,4,3,5,3,4,4,4,5,3,4,4,5,3,2,1,4,2,2,2,4,2,2,2,2,4,5,1,
%T 1,1,3,1,2,2,5,1,1,3,1,2,1,2,1,4,3,3,3,1,0,0,2,2,2,1,7,1,0,4,1,3,1,1,
%U 2,2,1,7,4,4,1,3,3,2,3,4,3,1,7,1,5,2,5,1,3,3,4,5,1,4,2,3,4,6,5,3
%N a(n) = |{0 < k < n-2: C(sigma(k) + phi(n-k)/2) - 1 is prime}|, where C(j) is the j-th Catalan number (A000108), sigma(k) is the sum of all positive divisors of k, and phi(.) is Euler's totient function.
%C It might seem that a(n) > 0 for all n > 63, but 9122 and 9438 are counterexamples.
%H Zhi-Wei Sun, <a href="/A235051/b235051.txt">Table of n, a(n) for n = 1..1800</a>
%e a(22) = 1 since sigma(8) + phi(14)/2 = 15 + 6/2 = 18 with C(18) - 1 = 477638699 prime.
%t sigma[n_]:=DivisorSigma[1,n]
%t f[n_,k_]:=CatalanNumber[sigma[k]+EulerPhi[n-k]/2]-1
%t a[n_]:=Sum[If[PrimeQ[f[n,k]],1,0],{k,1,n-3}]
%t Table[a[n],{n,1,100}]
%o (PARI) C(n)=binomial(2*n, n)/(n+1)
%o a(n)=sum(k=1,n-3,ispseudoprime(C(sigma(k)+eulerphi(n-k)/2)-1)) \\ _Charles R Greathouse IV_, Jan 03 2014
%Y Cf. A000010, A000040, A000108, A000203, A053427, A053429, A231885, A234963.
%K nonn
%O 1,6
%A _Zhi-Wei Sun_, Jan 02 2014
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