A236508 a(n) = |{0 < k < n-2: p = 2*phi(k) + phi(n-k)/2 - 1, p + 2, p + 6 and prime(p) + 6 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, 2, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1

Offset

1,11

Comments

Conjecture: a(n) > 0 for all n > 146.

We have verified this for n up to 52000.

The conjecture implies that there are infinitely many prime triples {p, p + 2, p + 6} with {prime(p), prime(p) + 6} a sexy prime pair. See A236509 for such primes p.

Links

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

Example

a(13) = 1 since 2*phi(3) + phi(10)/2 - 1 = 5, 5 + 2 = 7, 5 + 6 = 11 and prime(5) + 6 = 11 + 6 = 17 are all prime.
a(244) = 1 since 2*phi(153) + phi(244-153)/2 - 1 = 2*96 + 72/2 - 1 = 227, 227 + 2 = 229, 227 + 6 = 233 and prime(227) + 6 = 1433 + 6 = 1439 are all prime.

Mathematica

p[n_]:=PrimeQ[n]&&PrimeQ[n+2]&&PrimeQ[n+6]&&PrimeQ[Prime[n]+6]
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, A022004, A023201, A046117, A236464, A236472, A236509.

Sequence in context: A025838 A285813 A236480 * A364784 A239000 A105248

Adjacent sequences: A236505 A236506 A236507 * A236509 A236510 A236511

Keyword

nonn

Author

Zhi-Wei Sun, Jan 27 2014

Status

approved