A238504 Number of primes p <= n with pi(pi((p-1)*n)) prime, where pi(x) denotes the number of primes not exceeding x.

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

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

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000

Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2016.

Example

a(3) = 1 since 3 and pi(pi((3-1)*3)) = pi(pi(6)) = pi(3) = 2 are both prime.
a(20) = 1 since 3 and pi(pi((3-1)*20)) = pi(pi(40)) = pi(12) = 5 are both prime.
a(48) = 1 since 29 and pi(pi((29-1)*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

Zhi-Wei Sun, Mar 06 2014

Status

approved