

A237705


Number of primes p < n with pi(np) prime, where pi(.) is given by A000720.


14



0, 0, 0, 0, 1, 2, 2, 3, 2, 2, 2, 1, 2, 3, 2, 3, 3, 2, 3, 3, 2, 4, 4, 3, 3, 1, 1, 3, 3, 2, 2, 1, 2, 6, 6, 5, 5, 4, 3, 5, 5, 4, 5, 5, 4, 6, 6, 6, 6, 3, 3, 5, 5, 5, 5, 2, 2, 5, 5, 3, 4, 5, 4, 8, 8, 3, 3, 1, 2, 8
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OFFSET

1,6


COMMENTS

Conjecture: (i) a(n) > 0 for all n > 4, and a(n) = 1 only for n = 5, 12, 26, 27, 32, 68.
(ii) For any integer n > 5, there is a prime p <= n with pi(n+p) prime.
(iii) If n > 32, then pi((np)^2) is prime for some prime p < n. Also, for each n > 6 there is an odd prime p < 2*n with pi((n  (p1)/2)^2) prime.
(iv) Any integer n > 11 can be written as p + q with p and pi(q^2 + q + 1) both prime.
(v) Each integer n > 34 can be written as k + m with k and m positive integers such that pi(k^2) and pi(2*m^2) are both prime.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000


EXAMPLE

a(5) = 1 since 2 and pi(52) = pi(3) = 2 are both prime.
a(12) = 1 since 7 and pi(127) = pi(5) = 3 are both prime.
a(15) = 2 since 3 and pi(153) = pi(12) = 5 are both prime, and 11 and pi(1511) = pi(4) = 2 are both prime.
a(26) = 1 since 23 and pi(2623) = 2 are both prime.
a(27) = 1 since 23 and pi(2723) = 2 are both prime.
a(32) = 1 since 29 and pi(3229) = 2 are both prime.
a(68) = 1 since 37 and pi(6837) = pi(31) = 11 are both prime.


MATHEMATICA

q[n_]:=PrimeQ[PrimePi[n]]
a[n_]:=Sum[If[q[nPrime[k]], 1, 0], {k, 1, PrimePi[n1]}]
Table[a[n], {n, 1, 70}]


CROSSREFS

Cf. A000040, A000720, A237284, A237578, A237582.
Sequence in context: A254296 A248371 A237768 * A031284 A064569 A145989
Adjacent sequences: A237702 A237703 A237704 * A237706 A237707 A237708


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Feb 11 2014


STATUS

approved



