OFFSET
1,12
COMMENTS
Conjecture: (i) a(n) > 0 for all n > 14.
(ii) For any integer n > 12, there is a positive integer k < n such that p = prime(k) + phi(n-k), (p^2 - 1)/2 - prime(p) and (p^2 - 1)/4 - prime(p) are all prime.
Clearly, part (i) of the conjecture implies that there are infinitely many primes p with p^2 - 1 - prime(p) and (p^2 - 1)/2 - prime(p) both prime.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
EXAMPLE
a(10) = 1 since prime(5) + phi(5)/2 = 11 + 2 = 13, 13^2 - 1 - prime(13) = 168 - 41 = 127 and (13^2 - 1)/2 - prime(13) = 84 - 41 = 43 are all prime.
a(71) = 1 since prime(19) + phi(52)/2 = 67 + 12 = 79, 79^2 - 1 - prime(79) = 6240 - 401 = 5839 and (79^2 - 1)/2 - prime(79) = 3120 - 401 = 2719 are all prime.
MATHEMATICA
PQ[n_]:=n>0&&PrimeQ[n]
p[n_]:=PrimeQ[n]&&PQ[n^2-1-Prime[n]]&&PQ[(n^2-1)/2-Prime[n]]
f[n_, k_]:=Prime[k]+EulerPhi[n-k]/2
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 17 2014
STATUS
approved