

A238890


a(n) = {0 < k <= n: prime(k*n)  pi(k*n) is prime}, where pi(x) denotes the number of primes not exceeding x.


2



1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 4, 1, 2, 2, 3, 6, 1, 1, 4, 4, 1, 5, 3, 5, 5, 4, 5, 1, 2, 5, 7, 6, 5, 2, 2, 4, 4, 4, 10, 6, 5, 5, 4, 6, 8, 7, 5, 8, 5, 8, 5, 3, 5, 9, 6, 7, 2, 2, 4, 6, 7, 8, 11, 8, 8, 10, 6, 8, 10, 2, 5, 11, 7, 5, 10, 10, 8, 7, 9, 8
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OFFSET

1,2


COMMENTS

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 for no n > 28.
(ii) If n > 7 is not equal to 34, then prime(k*n) + pi(k*n) is prime for some k = 1, ..., n.
The conjecture implies that there are infinitely many primes p with p  pi(pi(p)) (or p + pi(pi(p)) prime.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..3000
ZhiWei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.


EXAMPLE

a(5) = 1 since prime(3*5)  pi(3*5) = 47  6 = 41 is prime.
a(28) = 1 since prime(18*28)  pi(18*28) = prime(504)  pi(504) = 3607  96 = 3511 is prime.


MATHEMATICA

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


CROSSREFS

Cf. A000040, A000720, A237578, A237712, A238573, A238576, A238878, A238881.
Sequence in context: A320107 A190321 A338409 * A266968 A237593 A338169
Adjacent sequences: A238887 A238888 A238889 * A238891 A238892 A238893


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Mar 06 2014


STATUS

approved



