A235924 a(n) = |{0 < k < n: p = phi(k) + phi(n-k)/3 + 1, q = prime(p) - p + 1 and r = prime(q) - q + 1 are all prime}|, where phi(.) is Euler's totient function.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 3, 0, 1, 0, 2, 3, 4, 3, 3, 1, 2, 3, 1, 6, 2, 9, 2, 5, 3, 4, 3, 8, 1, 4, 3, 9, 2, 3, 5, 6, 6, 7, 3, 8, 7, 6, 4, 4, 5, 7, 3, 6, 5, 1, 4, 6, 6, 2, 3, 4, 5, 4, 11, 4, 5, 4, 7, 2, 5, 5, 5, 2, 6, 2, 5, 5, 7

Offset

1,13

Comments

Conjecture: a(n) > 0 for all n > 37.

This implies that there are infinitely many primes p with q = prime(p) - p + 1 and r = prime(q) - q + 1 both prime.

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(6) + phi(14)/3 + 1 = 5, prime(5) - 4 = 11 - 4 = 7 and prime(7) - 6 = 17 - 6 = 11 are all prime.
a(77) = 1 since phi(59) + phi(18)/3 + 1 = 61, prime(61) - 60 = 283 - 60 = 223 and prime(223) - 222 = 1409 - 222 = 1187 are all prime.
a(1471) = 1 since phi(25) + phi(1446)/3 + 1 = 181, prime(181) - 180 = 1087 - 180 = 907 and prime(907) - 906 = 7057 - 906 = 6151 are all prime.

Mathematica

q[n_]:=Prime[n]-n+1
f[n_, k_]:=EulerPhi[k]+EulerPhi[n-k]/3+1
p[n_, k_]:=PrimeQ[f[n, k]]&&PrimeQ[q[f[n, k]]]&&PrimeQ[q[q[f[n, k]]]]
a[n_]:=Sum[If[p[n, k], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A000010, A000040, A234694, A234695.

Sequence in context: A082886 A287179 A236511 * A097304 A136745 A214157

Adjacent sequences: A235921 A235922 A235923 * A235925 A235926 A235927

Keyword

nonn

Author

Zhi-Wei Sun, Jan 17 2014

Status

approved