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

Author

Zhi-Wei Sun, Jan 07 2014

Keywords

• nonn

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.

Examples

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

Links

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

Crossrefs

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

Programs

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}]