A236417 a(n) = |{0 < k < n: p = phi(k)/2 + phi(n-k)/12 + 1 and A047967(p) are both prime}|.

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, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 2, 1, 0, 1, 0, 3, 1, 0, 1, 1, 1, 2, 1, 2, 0, 1, 2, 2, 2, 1, 2, 1, 1, 3, 1, 1, 4, 2, 0, 1, 3, 2, 2, 0, 2, 2, 4, 2, 3, 0, 3, 2

Offset: 1

Author

Zhi-Wei Sun, Jan 25 2014

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 98.

We have verified this for n up to 36000.

The conjecture implies that there are infinitely many primes p with A047967(p) prime.

Examples

			a(36) = 1 since phi(23)/2 + phi(13)/12 + 1 = 13 with A047967(13) = 83 prime.
a(71) = 1 since phi(43)/2 + phi(28)/12 + 1 = 23 with A047967(23) = 1151 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

Crossrefs

Cf. A000010, A000040, A047967.

Programs

Mathematica

pq[n_]:=PrimeQ[n]&&PrimeQ[PartitionsP[n]-PartitionsQ[n]]
f[n_,k_]:=EulerPhi[k]/2+EulerPhi[n-k]/12+1
a[n_]:=Sum[If[pq[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]