A234963 Number of ways to write n = k + m with k > 0 and m > 2 such that C(2*sigma(k) + phi(m), sigma(k) + phi(m)/2) - 1 is prime, where sigma(k) is the sum of all positive divisors of k and phi(.) is Euler's totient function.

0, 0, 0, 1, 1, 2, 3, 0, 3, 2, 2, 3, 3, 5, 3, 4, 3, 3, 3, 2, 3, 0, 3, 3, 4, 3, 0, 1, 2, 3, 1, 2, 3, 3, 1, 3, 3, 4, 1, 2, 3, 3, 2, 6, 4, 1, 4, 2, 3, 2, 2, 2, 4, 3, 2, 3, 3, 2, 4, 3, 3, 0, 2, 3, 1, 3, 1, 2, 0, 3, 1, 4, 4, 4, 1, 0, 5, 2, 1, 3, 2, 2, 1, 2, 1

Offset

1,6

Comments

Conjecture: a(n) > 0 for all n >= 180.

Clearly, this implies that there are infinitely many primes of the form C(2*n,n) - 1. We have verified the conjecture for n up to 10000.

Note that every n = 400, ..., 9123 can be written as k + m with k > 0 and m > 0 such that f(k, m) = sigma(k) + phi(m) is even and C(f(k, m) + 2, f(k, m)/2 + 1) + 1 is prime, but this fails for n = 9124.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..2000

Example

a(5) = 1 since 5 = 1 + 4 with C(2*sigma(1) + phi(4), sigma(1) + phi(4)/2) - 1 = C(4, 2) - 1 = 5 prime.
a(28) = 1 since 28 = 2 + 26 with C(2*sigma(2) + phi(26), sigma(2) + phi(26)/2) - 1 = C(18, 9) - 1 = 48619 prime.

Mathematica

sigma[n_] := DivisorSigma[1, n];
f[n_, k_] := Binomial[2*sigma[k] + EulerPhi[n-k], 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}]

Crossrefs

Cf. A000010, A000040, A000203, A000984, A066726, A066699, A232270, A233544, A234451, A234470, A234503, A234514, A234567, A234615.

Sequence in context: A319148 A193682 A051933 * A346275 A131900 A082116

Adjacent sequences: A234960 A234961 A234962 * A234964 A234965 A234966

Keyword

nonn

Author

Zhi-Wei Sun, Jan 01 2014

Status

approved