OFFSET
1,8
COMMENTS
Conjecture: a(n) > 0 for all n > 31.
This implies that there are infinitely many primes p with {prime(p) - p - 1, prime(p) - p + 1} a twin prime pair.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014
EXAMPLE
a(20) = 1 since phi(2) + phi(18)/2 + 1 = 5, prime(5) - 5 - 1 = 5 and prime(5) - 5 + 1 = 7 are all prime.
a(36) = 1 since phi(21) + phi(15)/2 + 1 = 17, prime(17) - 17 - 1 = 41 and prime(17) - 17 + 1 = 43 are all prime.
MATHEMATICA
p[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]-n-1]&&PrimeQ[Prime[n]-n+1]
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 19 2014
STATUS
approved