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
Keywords
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.
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}]
Comments