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

Author

Zhi-Wei Sun, Feb 28 2014

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 3, 4, 8, 10, 24, 41.

(ii) For any integer n > 6, there is a positive integer k < n with n^2 + pi(k*n) - 1 prime.

(iii) If n > 2, then pi(n^2) - pi(k*n) is prime for some 0 < k < n. If n > 1, then pi(n^2) + pi(k*n) - 1 is prime for some 0 < k < n.

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.

Crossrefs

Cf. A000040, A000720, A237578, A237615, A237712, A238570.

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}]