A237657 a(n) = |{n < m < 2*n: pi(m) and pi(m^2) are both prime}|, where pi(.) is given by A000720.

0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 5, 5, 4, 3, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 6, 6, 6, 5, 4, 4, 4, 4, 5, 5, 5, 5, 5, 4, 4

Offset

1,18

Comments

Conjecture: (i) a(n) > 0 for all n > 8.

(ii) For any integer n > 1 there is a prime p <= n such that n + pi(p) is prime. Also, for n > 5 there is a prime p with n < p < 2*n such that pi(p) is prime.

(iii) For each n > 20, there is a prime p with n < p < 2*n such that pi(p^2) is prime.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Example

a(4) = 1 since pi(6) = 3 and pi(6^2) = 11 are both prime.
a(10) = 1 since pi(17) = 7 and pi(17^2) = 61 are both prime.
a(17) = 1 since pi(33) = 11 and pi(33^2) = 181 are both prime.

Mathematica

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

Crossrefs

Cf. A000040, A000720, A038107, A237578, A237643, A237656.

Sequence in context: A064099 A134021 A330558 * A244317 A130255 A082527

Adjacent sequences: A237654 A237655 A237656 * A237658 A237659 A237660

Keyword

nonn

Author

Zhi-Wei Sun, Feb 10 2014

Status

approved