A236097 a(n) = |{0 < k < n-2: p = phi(k) + phi(n-k)/2 + 1, prime(p) - p - 1 and prime(p) - p + 1 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, 3, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 2, 0, 5, 5, 2, 4, 1, 5, 3, 3, 2, 4, 4, 9, 5, 9, 4, 10, 3, 6, 6, 8, 5, 10, 4, 4, 7, 8, 10, 5, 8, 9, 9, 4, 11, 3, 5, 5, 9, 5, 4, 4, 5, 6, 8, 7, 6, 3, 11, 4, 8, 10, 9, 8, 7, 6, 11, 7, 9, 4, 6, 5, 6, 2, 9, 4, 7, 6, 7, 10, 9

Offset

1,8

Comments

Conjecture: a(n) > 0 for all n > 31.

This implies that there are infinitely many primes p with {prime(p) - p - 1, prime(p) - p + 1} a twin prime pair.

Links

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

Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Example

a(20) = 1 since phi(2) + phi(18)/2 + 1 = 5, prime(5) - 5 - 1 = 5 and prime(5) - 5 + 1 = 7 are all prime.
a(36) = 1 since phi(21) + phi(15)/2 + 1 = 17, prime(17) - 17 - 1 = 41 and prime(17) - 17 + 1 = 43 are all prime.

Mathematica

p[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]-n-1]&&PrimeQ[Prime[n]-n+1]
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, A014574, A234694, A234695, A235924, A236074, A236119.

Sequence in context: A264033 A236293 A056044 * A239319 A236468 A116685

Adjacent sequences: A236094 A236095 A236096 * A236098 A236099 A236100

Keyword

nonn

Author

Zhi-Wei Sun, Jan 19 2014

Status

approved