OFFSET
1,19
COMMENTS
Conjecture: a(n) > 0 for all n > 357.
This is much stronger than the twin prime conjecture. Actually it implies that there are infinitely many primes p such that {p, p + 2} and {prime(p+2), prime(p+2) + 2} are both twin prime pairs. See A236457 for such primes p.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
EXAMPLE
a(18) = 1 since 18 = 3 + 15 with phi(3) + phi(15)/4 - 1 = 3, 3 + 2 = 5 and prime(5) + 2 = 13 all prime.
a(50) = 1 since 50 = 16 + 34 with phi(16) + phi(34)/4 - 1 = 11, 11 + 2 = 13 and prime(13) + 2 = 43 all prime.
a(929) = 1 since 929 = 441 + 488 with phi(441) + phi(488)/4 - 1 = 252 + 60 - 1 = 311, 311 + 2 = 313 and prime(313) + 2 = 2083 all prime.
MATHEMATICA
p[n_]:=PrimeQ[n]&&PrimeQ[n+2]&&PrimeQ[Prime[n+2]+2]
f[n_, k_]:=EulerPhi[k]+EulerPhi[n-k]/4-1
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 26 2014
STATUS
approved