OFFSET
1,8
COMMENTS
Conjecture: a(n) > 0 for every n = 250, 251, ....
This implies that there are infinitely many twin prime pairs {p, p + 2} with {prime(p) - 2, prime(p)} also a twin prime pair. It is stronger than the twin prime conjecture.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
EXAMPLE
a(33) = 1 since 33 = 7 + 26 with phi(7) + phi(26)/2 - 1 = 11, 11 + 2 = 13 and prime(11) - 2 = 31 - 2 = 29 all prime.
a(278) = 1 since 278 = 61 + 217 with phi(61) + phi(217)/2 - 1 = 60 + 90 - 1 = 149, 149 + 2 = 151 and prime(149) - 2 = 859 - 2 = 857 all prime.
MATHEMATICA
p[n_]:=PrimeQ[n]&&PrimeQ[n+2]&&PrimeQ[Prime[n]-2]
f[n_, k_]:=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