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

Author

Zhi-Wei Sun, Jan 26 2014

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

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

Programs

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}]