OFFSET
1,10
COMMENTS
Conjecture: a(n) > 0 for all n > 410.
This implies that there are infinitely many positive integers m with C(2*m, m) - prime(m) prime. We have verified the conjecture for n up to 51000.
See A236248 for a list of known numbers m with C(2*m, m) - prime(m) prime.
See also A236249 for those primes of the form C(2*m, m) - prime(m).
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
EXAMPLE
a(12) = 1 since phi(2) + phi(10)/4 = 1 + 1 = 2 with C(2*2, 2) - prime(2) = 6 - 3 = 3 prime.
a(33) = 1 since phi(1) + phi(32)/4 = 1 + 4 = 5 with C(2*5, 5) - prime(5) = 252 - 11 = 241 prime.
a(697) = 1 since phi(452) + phi(697-452)/4 = 224 + 42 = 266 with C(2*266, 266) - prime(266) = C(532, 266) - 1699 prime.
MATHEMATICA
p[n_]:=IntegerQ[n]&&PrimeQ[Binomial[2n, n]-Prime[n]]
f[n_, k_]:=EulerPhi[k]+EulerPhi[n-k]/4
a[n_]:=Sum[If[p[f[n, k]], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 21 2014
STATUS
approved