

A238504


Number of primes p <= n with pi(pi((p1)*n)) prime, where pi(x) denotes the number of primes not exceeding x.


2



0, 0, 1, 1, 2, 2, 2, 3, 2, 3, 2, 1, 3, 2, 4, 3, 1, 2, 3, 1, 1, 2, 1, 2, 2, 1, 1, 4, 1, 2, 4, 4, 5, 5, 3, 4, 4, 6, 4, 2, 1, 1, 3, 4, 6, 4, 4, 1, 3, 3, 2, 4, 6, 2, 2, 3, 3, 3, 4, 10, 5, 6, 7, 5, 4, 5, 5, 3, 5, 6, 6, 5, 5, 4, 5, 6, 6, 4, 3, 4
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OFFSET

1,5


COMMENTS

Conjecture: (i) a(n) > 0 for all n > 2.
(ii) If n > 92, then pi(p*n) is prime for some prime p <= n. If n > 39, then pi(pi(p*n)) is prime for some p <= n.
See also A238902 for another conjecture involving pi(pi(x)).


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..5000
ZhiWei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.


EXAMPLE

a(3) = 1 since 3 and pi(pi((31)*3)) = pi(pi(6)) = pi(3) = 2 are both prime.
a(20) = 1 since 3 and pi(pi((31)*20) = pi(pi(40)) = pi(12) = 5 are both prime.
a(48) = 1 since 29 and pi(pi((291)*48) = pi(pi(1344)) = pi(217) = 47 are both prime.


MATHEMATICA

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


CROSSREFS

Cf. A000040, A000720, A237578, A238890, A238902.
Sequence in context: A058013 A223934 A237531 * A031356 A304522 A308641
Adjacent sequences: A238501 A238502 A238503 * A238505 A238506 A238507


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Mar 06 2014


STATUS

approved



