A236439 a(n) = |{0 < k < n-2: A000009(m)^2 + A047967(m)^2 is prime with m = k + phi(n-k)/2}|, where phi(.) is Euler's totient function.

0, 0, 0, 1, 2, 3, 3, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 4, 2, 3, 2, 3, 5, 4, 3, 2, 6, 6, 4, 2, 1, 8, 4, 4, 3, 1, 6, 4, 3, 3, 3, 3, 3, 4, 4, 5, 3, 4, 5, 3, 3, 7, 4, 5, 5, 5, 11, 7, 6, 3, 7, 8, 6, 5, 5, 8, 6, 7, 11, 7, 5, 7, 8, 7, 7, 5, 10, 10, 5, 6, 8, 6, 10, 8, 6, 8, 11, 10, 6, 10, 7, 7, 9, 4, 9, 11, 8, 13, 7

Offset: 1

Author

Zhi-Wei Sun, Jan 25 2014

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 3.

We have verified this for n up to 50000.

The conjecture implies that there are infinitely many positive integers m with A000009(m)^2 + A047967(m)^2 prime. See A236440 for such numbers m.

Examples

			a(14) = 1 since 2 + phi(12)/2 = 4 with A000009(4)^2 + A047967(4)^2 = 2^2 + 3^2 = 13 prime.
a(17) = 1 since 10 + phi(7)/2 = 13 with A000009(13)^2 + A047967(13)^2 = 18^2 + 83^2 = 7213 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. A000009, A000010, A000040, A047967, A236412, A236417, A236419, A236440.

Programs

Mathematica

p[n_]:=PrimeQ[PartitionsQ[n]^2+(PartitionsP[n]-PartitionsQ[n])^2]
a[n_]:=Sum[If[p[k+EulerPhi[n-k]/2],1,0],{k,1,n-3}]
Table[a[n],{n,1,100}]