A237595 a(n) = |{1 <= k <= n: n + pi(k^2) is prime}|, where pi(.) is given by A000720.

0, 1, 3, 0, 3, 1, 3, 3, 3, 1, 5, 2, 6, 3, 4, 2, 6, 3, 7, 3, 2, 6, 8, 1, 10, 3, 5, 8, 9, 2, 9, 6, 3, 5, 14, 5, 11, 6, 9, 3, 13, 8, 11, 8, 8, 6, 8, 8, 11, 9, 6, 12, 15, 10, 11, 5, 11, 12, 13, 9, 12, 9, 5, 17, 15, 9, 18, 13, 11, 12

Offset: 1

Author

Zhi-Wei Sun, Feb 09 2014

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 4.

(ii) If n > 1 then n + pi(k*(k-1)) is prime for some k = 1, ..., n.

(iii) For any integer n > 1, there is a positive integer k <= (n+1)/2 such that pi(n + k*(k+1)/2) is prime.

(iv) Any integer n > 1 can be written as p + pi(k*(k+1)/2), where p is a prime and k is among 1, ..., n-1.

Examples

			a(2) = 1 since 2 + pi(1^2) = 2 is prime.
a(6) = 1 since 6 + pi(6^2) = 6 + 11 = 17 is prime.
a(10) = 1 since 10 + pi(5^2) = 10 + 9 = 19 is prime.
a(21) = 2 since 21 + pi(2^2) = 23 and 21 + pi(9^2) = 43 are both prime.
a(24) = 1 since 24 + pi(21^2) = 24 + 85 = 109 is prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..3000
• Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Crossrefs

Cf. A000040, A000720, A237453, A237496, A237497, A237578, A237582.

Programs

Mathematica

a[n_]:=Sum[If[PrimeQ[n+PrimePi[k^2]],1,0],{k,1,n}]
Table[a[n],{n,1,70}]