A236480 a(n) = |{0 < k < n-2: p = 2*phi(k) + phi(n-k)/2 + 1, prime(p) + 2 and prime(prime(p)) + 2 are all prime}|, where phi(.) is Euler's totient function.

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 2, 1, 1, 2, 1, 3, 2, 2, 0, 2, 3, 1, 2, 1, 3, 3, 2, 2, 1, 1, 1, 3, 0, 2, 3, 2, 1, 3, 0, 2, 0, 1, 1, 1, 1, 2, 0, 0, 0, 0, 2, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

1,11

Comments

Conjecture: a(n) > 0 for every n = 640, 641, ....

We have verified this for n up to 75000.

The conjecture implies that there are infinitely many primes p with prime(p) + 2 and prime(prime(p)) + 2 both prime.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Example

a(8) = 1 since 2*phi(3) + phi(5)/2 + 1 = 7, prime(7) + 2 = 17 + 2 = 19 and prime(prime(7)) + 2 = prime(17) + 2 = 61 are all prime.
a(667) = 1 since 2*phi(193) + phi(667-193)/2 + 1 = 384 + 78 + 1 = 463, prime(463) + 2 = 3299 + 2 = 3301 and prime(prime(463)) + 2 = prime(3299) + 2 = 30559 are all prime.

Mathematica

p[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]+2]&&PrimeQ[Prime[Prime[n]]+2]
f[n_, k_]:=2*EulerPhi[k]+EulerPhi[n-k]/2+1
a[n_]:=Sum[If[p[f[n, k]], 1, 0], {k, 1, n-3}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A000010, A000040, A001359, A006512, A236456, A236457, A236458, A236468, A236470, A236481.

Sequence in context: A088428 A025838 A285813 * A236508 A364784 A239000

Adjacent sequences: A236477 A236478 A236479 * A236481 A236482 A236483

Keyword

nonn

Author

Zhi-Wei Sun, Jan 26 2014

Status

approved