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A235051
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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.
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1
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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, 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, 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
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OFFSET
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1,6
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COMMENTS
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It might seem that a(n) > 0 for all n > 63, but 9122 and 9438 are counterexamples.
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LINKS
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EXAMPLE
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a(22) = 1 since sigma(8) + phi(14)/2 = 15 + 6/2 = 18 with C(18) - 1 = 477638699 prime.
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MATHEMATICA
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sigma[n_]:=DivisorSigma[1, n]
f[n_, k_]:=CatalanNumber[sigma[k]+EulerPhi[n-k]/2]-1
a[n_]:=Sum[If[PrimeQ[f[n, k]], 1, 0], {k, 1, n-3}]
Table[a[n], {n, 1, 100}]
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PROG
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(PARI) C(n)=binomial(2*n, n)/(n+1)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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