A238890 a(n) = |{0 < k <= n: prime(k*n) - pi(k*n) is prime}|, where pi(x) denotes the number of primes not exceeding x.

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

Offset: 1

Author

Zhi-Wei Sun, Mar 06 2014

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 for no n > 28.

(ii) If n > 7 is not equal to 34, then prime(k*n) + pi(k*n) is prime for some k = 1, ..., n.

The conjecture implies that there are infinitely many primes p with p - pi(pi(p)) (or p + pi(pi(p))) prime.

Examples

			a(5) = 1 since prime(3*5) - pi(3*5) = 47 - 6 = 41 is prime.
a(28) = 1 since prime(18*28) - pi(18*28) = prime(504) - pi(504) = 3607 - 96 = 3511 is prime.
		

Links

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

Crossrefs

Cf. A000040, A000720, A237578, A237712, A238573, A238576, A238878, A238881.

Programs

Mathematica

p[k_]:=PrimeQ[Prime[k]-PrimePi[k]]
a[n_]:=Sum[If[p[k*n],1,0],{k,1,n}]
Table[a[n],{n,1,80}]