OFFSET
1,32
COMMENTS
Conjecture: a(n) > 0 for all n > 211.
This implies that there are infinitely many primes p with {prime(p), prime(p) + 4, prime(p) + 6} a prime triple. See A236462 for such primes p.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
EXAMPLE
a(30) = 1 since 30 = 13 + 17 with phi(13) + phi(17)/2 - 1 = 19, prime(19) + 4 = 67 + 4 = 71 and prime(19) + 6 = 73 all prime.
a(831) = 1 since 831 = 66 + 765 with phi(66) + phi(765)/2 - 1 = 20 + 192 - 1 = 211, prime(211) + 4 = 1297 + 4 = 1301 and prime(211) + 6 = 1303 all prime.
MATHEMATICA
p[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]+4]&&PrimeQ[Prime[n]+6]
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