OFFSET
1,13
COMMENTS
Conjecture: (i) a(n) > 0 for all n > 12.
(ii) Any even number greater than 4 can be written as p + q with p, q, phi(p+2) - 1 and phi(p+2) + 1 all prime.
Part (i) implies both Lemoine's conjecture (cf. A046927) and the twin prime conjecture, while part (ii) unifies Goldbach's conjecture and the twin prime conjecture.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Unification of Goldbach's conjecture and the twin prime conjecture, a message to Number Theory List, Jan. 29, 2014.
EXAMPLE
a(9) = 1 since 2*9 - 1 = 2*7 + 3 with 7, 3, phi(7+1) - 1 = 3 and phi(7+1) + 1 = 5 all prime.
a(934) = 1 since 2*934 - 1 = 2*457 + 953 with 457, 953, phi(457+1) - 1 = 227 and phi(457+1) + 1 = 229 all prime.
MATHEMATICA
PQ[n_]:=PrimeQ[EulerPhi[n]-1]&&PrimeQ[EulerPhi[n]+1]
a[n_]:=Sum[If[PQ[Prime[k]+1]&&PrimeQ[2n-1-2*Prime[k]], 1, 0], {k, 1, PrimePi[n-1]}]
Table[a[n], {n, 1, 70}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 04 2014
STATUS
approved