A235051 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.

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

Offset

1,6

Comments

It might seem that a(n) > 0 for all n > 63, but 9122 and 9438 are counterexamples.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..1800

Example

a(22) = 1 since sigma(8) + phi(14)/2 = 15 + 6/2 = 18 with C(18) - 1 = 477638699 prime.

Mathematica

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}]

Prog

(PARI) C(n)=binomial(2*n, n)/(n+1)
a(n)=sum(k=1, n-3, ispseudoprime(C(sigma(k)+eulerphi(n-k)/2)-1)) \\ Charles R Greathouse IV, Jan 03 2014

Crossrefs

Cf. A000010, A000040, A000108, A000203, A053427, A053429, A231885, A234963.

Sequence in context: A309255 A201443 A291762 * A322814 A324399 A325382

Adjacent sequences: A235048 A235049 A235050 * A235052 A235053 A235054

Keyword

nonn

Author

Zhi-Wei Sun, Jan 02 2014

Status

approved