A239430 Number of ways to write n = k + m with k > 0 and m > 0 such that pi(2*k) - pi(k) is prime and pi(2*m) - pi(m) is a square, where pi(x) denotes the number of primes not exceeding x.

0, 0, 0, 0, 1, 1, 2, 2, 4, 3, 3, 3, 2, 4, 2, 5, 3, 4, 5, 1, 5, 3, 6, 7, 5, 9, 3, 7, 5, 4, 7, 5, 9, 5, 5, 4, 2, 4, 2, 5, 4, 6, 7, 5, 9, 6, 9, 8, 7, 10, 8, 10, 6, 7, 6, 6, 7, 6, 5, 6, 7, 5, 5, 6, 7, 8, 7, 10, 11, 12, 11, 7, 6, 9, 10, 8, 7, 6, 7, 5

Offset: 1

Author

Zhi-Wei Sun, Mar 20 2014

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 4, and a(n) = 1 only for n = 5, 6, 20.

Examples

			a(5) = 1 since 5 = 4 + 1 with pi(2*4) - pi(4) = 4 - 2 = 2 prime and pi(2*1) - pi(1) = 1^2.
a(20) = 1 since 20 = 8 + 12 with pi(2*8) - pi(8) = 6 - 4 = 2 prime and pi(2*12) - pi(12) = 9 - 5 = 2^2.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Problems on combinatorial properties of primes, preprint, arXiv:1402.6641, 2014.

Crossrefs

Cf. A000040, A000290, A000720, A239428.

Programs

Mathematica

SQ[n_]:=IntegerQ[Sqrt[n]]
s[n_]:=SQ[PrimePi[2n]-PrimePi[n]]
p[n_]:=PrimeQ[PrimePi[2n]-PrimePi[n]]
a[n_]:=Sum[If[p[k]&&s[n-k],1,0],{k,1,n-1}]
Table[a[n],{n,1,80}]