OFFSET
1,11
COMMENTS
Conjecture: a(n) > 0 for every n = 640, 641, ....
We have verified this for n up to 75000.
The conjecture implies that there are infinitely many primes p with prime(p) + 2 and prime(prime(p)) + 2 both prime.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
EXAMPLE
a(8) = 1 since 2*phi(3) + phi(5)/2 + 1 = 7, prime(7) + 2 = 17 + 2 = 19 and prime(prime(7)) + 2 = prime(17) + 2 = 61 are all prime.
a(667) = 1 since 2*phi(193) + phi(667-193)/2 + 1 = 384 + 78 + 1 = 463, prime(463) + 2 = 3299 + 2 = 3301 and prime(prime(463)) + 2 = prime(3299) + 2 = 30559 are all prime.
MATHEMATICA
p[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]+2]&&PrimeQ[Prime[Prime[n]]+2]
f[n_, k_]:=2*EulerPhi[k]+EulerPhi[n-k]/2+1
a[n_]:=Sum[If[p[f[n, k]], 1, 0], {k, 1, n-3}]
Table[a[n], {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 26 2014
STATUS
approved