A236472 a(n) = |{0 < k < n: p = prime(k) + phi(n-k), prime(p) + 2 and prime(p) + 6 are all prime}|, where phi(.) is Euler's totient function.

0, 1, 1, 0, 1, 2, 2, 2, 2, 1, 2, 1, 2, 1, 0, 1, 1, 2, 3, 0, 1, 1, 1, 2, 0, 1, 1, 0, 0, 2, 0, 2, 2, 1, 0, 0, 3, 1, 2, 0, 2, 2, 2, 1, 0, 0, 4, 1, 0, 0, 0, 0, 5, 0, 1, 1, 1, 2, 1, 1, 3, 0, 0, 2, 2, 0, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 3, 3, 3, 2, 0, 0, 2, 1, 1, 3, 0, 0, 2, 0, 3, 0, 0, 1, 1, 0, 2, 0, 0

Offset

1,6

Comments

Conjecture: a(n) > 0 for every n = 330, 331, ....

We have verified this for n up to 80000.

The conjecture implies that there are infinitely many prime triples of the form {prime(p), prime(p) + 2, prime(p) + 6} with p prime. See A236464 for such primes p.

Links

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

Example

a(10) = 1 since prime(2) + phi(8) = 3 + 4 = 7, prime(7) + 2 = 17 + 2 = 19 and prime(7) + 6 = 23 are all prime.
a(877) = 1 since prime(784) + phi(877-784) = 6007 + 60 = 6067, prime(6067) + 2 = 60101 + 2 = 60103 and prime(6067) + 6 = 60107 are all prime.

Mathematica

p[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]+2]&&PrimeQ[Prime[n]+6]
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}]

Crossrefs

Cf. A000010, A000040, A022004, A236456, A236460, A236464, A236467, A236470.

Sequence in context: A336013 A172363 A181877 * A175357 A232800 A248380

Adjacent sequences: A236469 A236470 A236471 * A236473 A236474 A236475

Keyword

nonn

Author

Zhi-Wei Sun, Jan 26 2014

Status

approved