A238568 a(n) = |{0 < k < n: n^2 - pi(k*n) is prime}|, where pi(x) denotes the number of primes not exceeding x.
0, 1, 1, 1, 2, 2, 2, 1, 2, 1, 3, 2, 4, 3, 4, 2, 2, 5, 5, 3, 4, 4, 8, 1, 3, 3, 4, 3, 4, 3, 6, 3, 4, 4, 3, 4, 6, 3, 5, 2, 1, 8, 3, 10, 6, 5, 5, 9, 7, 6, 3, 8, 7, 9, 2, 5, 5, 2, 2, 9, 7, 3, 5, 8, 7, 6, 8, 7, 9, 9, 6, 3, 7, 8, 14, 5, 9, 10, 8, 11
Offset: 1
Keywords
Examples
a(2) = 1 since 2^2 - pi(1*2) = 4 - 1 = 3 is prime. a(3) = 1 since 3^2 - pi(1*3) = 9 - 2 = 7 is prime. a(4) = 1 since 4^2 - pi(3*4) = 16 - 5 = 11 is prime. a(8) = 1 since 8^2 - pi(4*8) = 64 - 11 = 53 is prime. a(10) = 1 since 10^2 - pi(6*10) = 100 - 17 = 83 is prime. a(24) = 1 since 24^2 - pi(14*24) = 576 - 67 = 509 is prime. a(41) = 1 since 41^2 - pi(10*41) = 1681 - 80 = 1601 is prime.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 1..4000
- Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2016.
Programs
-
Mathematica
p[k_,n_]:=PrimeQ[n^2-PrimePi[k*n]] a[n_]:=Sum[If[p[k,n],1,0],{k,1,n-1}] Table[a[n],{n,1,80}]
Comments