A236619 a(n) = |{0 < k < n: prime(m)^3 + 2*m^3 and m^3 + 2*prime(m)^3 are both prime with m = 3*phi(k) + phi(n-k) - 1}|, where phi(.) is Euler's totient function.

0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 1, 2, 0, 0, 2, 2, 1, 1, 0, 1, 1, 2, 2, 1, 2, 4, 0, 1, 3, 0, 2, 3, 3, 2, 3, 1, 3, 2, 3, 3, 2, 4, 3, 4, 2, 0, 2, 5, 4, 2, 4, 2, 2, 3, 5, 5, 6

Offset

1,3

Comments

Conjecture: a(n) > 0 for every n = 90, 91, ....

We have verified this for n up to 100000.

The conjecture implies that there are infinitely many positive integers m with prime(m)^3 + 2*m^3 and m^3 + 2*prime(m)^3 both prime.

Links

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

Example

a(51) = 1 since 3*phi(35) + phi(51-35) - 1 = 3*24 + 8 - 1 = 79 with prime(79)^3 + 2*79^3 = 401^3 + 2*79^3 = 65467279 and 79^3 + 2*prime(79)^3 = 79^3 + 2*401^3 = 129455441 both prime.

Mathematica

p[n_]:=PrimeQ[Prime[n]^3+2*n^3]&&PrimeQ[n^3+2*Prime[n]^3]
f[n_, k_]:=3*EulerPhi[k]+EulerPhi[n-k]-1
a[n_]:=Sum[If[p[f[n, k]], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A000010, A000040, A000578, A173587, A220413, A236192, A236574.

Sequence in context: A033322 A329679 A130713 * A355627 A300828 A297246

Adjacent sequences: A236616 A236617 A236618 * A236620 A236621 A236622

Keyword

nonn

Author

Zhi-Wei Sun, Jan 29 2014

Status

approved