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A238504 Number of primes p <= n with pi(pi((p-1)*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 (list; graph; refs; listen; history; text; internal format)
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, 2014.

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

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Last modified June 17 19:10 EDT 2019. Contains 324198 sequences. (Running on oeis4.)