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A236470
a(n) = |{0 < k < n: p = prime(k) + phi(n-k), p + 2 and prime(p) + 2 are all prime}|, where phi(.) is Euler's totient function.
6
0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 2, 1, 2, 1, 1, 1, 1, 0, 2, 2, 2, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 1, 3, 0, 1, 1, 1, 0, 1, 0, 0, 1, 2, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 2, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0
OFFSET
1,14
COMMENTS
Conjecture: a(n) > 0 for all n > 948.
We have verified this for n up to 50000.
The conjecture implies that there are infinitely many primes p with p + 2 and prime(p) + 2 both prime. See A236458 for such primes p.
EXAMPLE
a(12) = 1 since prime(5) + phi(7) = 11 + 6 = 17, 17 + 2 = 19 and prime(17) + 2 = 59 + 2 = 61 are all prime.
a(97) = 1 since prime(7) + phi(90) = 17 + 24 = 41, 41 + 2 = 43 and prime(41) + 2 = 179 + 2 = 181 are all prime.
MATHEMATICA
p[n_]:=PrimeQ[n]&&PrimeQ[n+2]&&PrimeQ[Prime[n]+2]
f[n_, k_]:=Prime[k]+EulerPhi[n-k]
a[n_]:=Sum[If[p[f[n, k]], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 100}]
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 26 2014
STATUS
approved