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A237657 a(n) = |{n < m < 2*n: pi(m) and pi(m^2) are both prime}|, where pi(.) is given by A000720. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 18 15:48 EDT 2020. Contains 337169 sequences. (Running on oeis4.)