A236468 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that p = phi(k) + phi(m)/2 - 1, p + 2 and prime(p) - 2 are all prime, where phi(.) is Euler's totient function.

0, 0, 0, 0, 0, 0, 0, 2, 2, 1, 3, 1, 1, 2, 2, 4, 0, 1, 2, 2, 1, 2, 1, 1, 2, 0, 3, 2, 2, 3, 4, 2, 1, 2, 5, 3, 4, 0, 6, 6, 1, 3, 1, 5, 4, 5, 2, 5, 1, 7, 1, 3, 2, 5, 1, 4, 1, 7, 0, 5, 4, 1, 8, 1, 5, 5, 1, 2, 5, 4, 4, 4, 4, 1, 5, 1, 7, 3, 3, 2, 2, 1, 8, 3, 3, 2, 2, 2, 6, 3, 7, 2, 6, 5, 1, 1, 5, 4, 9, 3

Offset

1,8

Comments

Conjecture: a(n) > 0 for every n = 250, 251, ....

This implies that there are infinitely many twin prime pairs {p, p + 2} with {prime(p) - 2, prime(p)} also a twin prime pair. It is stronger than the twin prime conjecture.

Links

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

Example

 a(33) = 1 since 33 = 7 + 26 with phi(7) + phi(26)/2 - 1 = 11, 11 + 2 = 13 and prime(11) - 2 = 31 - 2 = 29 all prime.
a(278) = 1 since 278 = 61 + 217 with phi(61) + phi(217)/2 - 1 = 60 + 90 - 1 = 149, 149 + 2 = 151 and prime(149) - 2 = 859 - 2 = 857 all prime.

Mathematica

p[n_]:=PrimeQ[n]&&PrimeQ[n+2]&&PrimeQ[Prime[n]-2]
f[n_, k_]:=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, A236097, A236456, A236460, A236467.

Sequence in context: A056044 A236097 A239319 * A116685 A355522 A268190

Adjacent sequences: A236465 A236466 A236467 * A236469 A236470 A236471

Keyword

nonn

Author

Zhi-Wei Sun, Jan 26 2014

Status

approved