A235358 a(n) = |{0 < k < n: g(n,k) - 1, g(n,k) + 1 and q(g(n,k)) - 1 are all prime with g(n,k) = phi(k) + phi(n-k)/8}|, where phi(.) is Euler's totient function and q(.) is the strict partition function (A000009).

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

Offset

1,42

Comments

Conjecture: a(n) > 0 for all n > 1234.

See also part (ii) of the conjecture in A235343.

We have verified the conjecture for n up to 100000.

Links

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

Example

a(50) = 1 since phi(10) + phi(40)/4 = 6 with 6 - 1, 6 + 1 and q(6) - 1 = 3 all prime.

Mathematica

f[n_, k_]:=EulerPhi[k]+EulerPhi[n-k]/8
p[n_, k_]:=PrimeQ[f[n, k]-1]&&PrimeQ[f[n, k]+1]&&PrimeQ[PartitionsQ[f[n, k]]-1]
a[n_]:=Sum[If[p[n, k], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A000009, A000010, A000040, A001359, A006512, A014574, A234514, A234567, A234615, A235343, A235344, A235346, A235356, A235357.

Sequence in context: A056931 A139569 A201590 * A086249 A176784 A176511

Adjacent sequences: A235355 A235356 A235357 * A235359 A235360 A235361

Keyword

nonn

Author

Zhi-Wei Sun, Jan 07 2014

Status

approved