A236470 a(n) = |{0 < k < n: p = prime(k) + phi(n-k), p + 2 and prime(p) + 2 are all prime}|, where phi(.) is Euler's totient function.

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

Offset

1,14

Comments

Conjecture: a(n) > 0 for all n > 948.

We have verified this for n up to 50000.

The conjecture implies that there are infinitely many primes p with p + 2 and prime(p) + 2 both prime. See A236458 for such primes p.

Links

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

Example

 a(12) = 1 since prime(5) + phi(7) = 11 + 6 = 17, 17 + 2 = 19 and prime(17) + 2 = 59 + 2 = 61 are all prime.
a(97) = 1 since prime(7) + phi(90) = 17 + 24 = 41, 41 + 2 = 43 and prime(41) + 2 = 179 + 2 = 181 are all prime.

Mathematica

p[n_]:=PrimeQ[n]&&PrimeQ[n+2]&&PrimeQ[Prime[n]+2]
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, A001359, A006512, A236097, A236456, A236458, A236468.

Sequence in context: A218854 A172303 A064391 * A206589 A086011 A124760

Adjacent sequences: A236467 A236468 A236469 * A236471 A236472 A236473

Keyword

nonn

Author

Zhi-Wei Sun, Jan 26 2014

Status

approved