OFFSET
1,5
COMMENTS
Conjecture: a(n) > 0 for all n > 84.
Clearly, this implies that there are infinitely many primes p with prime(p) - p + 1 and p*(p+1) - prime(p) both prime.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
EXAMPLE
a(10) = 1 since 10 = 1 + 9 with phi(1) + phi(9)/2 + 1 = 5, prime(5) - 5 + 1 = 7 and 5*6 - prime(5) = 19 all prime.
a(95) = 1 since 95 = 62 + 33 with phi(62) + phi(33)/2 + 1 = 41, prime(41) - 41 + 1 = 139 and 41*42 - prime(41) = 1543 all prime.
a(421) = 1 since 421 = 289 + 132 with phi(289) + phi(132)/2 + 1 = 293, prime(293) - 293 + 1 = 1621 and 293*294 - prime(293) = 84229 all prime.
MATHEMATICA
PQ[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]-n+1]&&PrimeQ[n(n+1)-Prime[n]]
f[n_, k_]:=EulerPhi[k]+EulerPhi[n-k]/2+1
a[n_]:=Sum[If[PQ[f[n, k]], 1, 0], {k, 1, n-3}]
Table[a[n], {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 13 2014
STATUS
approved