

A238568


a(n) = {0 < k < n: n^2  pi(k*n) is prime}, where pi(x) denotes the number of primes not exceeding x.


2



0, 1, 1, 1, 2, 2, 2, 1, 2, 1, 3, 2, 4, 3, 4, 2, 2, 5, 5, 3, 4, 4, 8, 1, 3, 3, 4, 3, 4, 3, 6, 3, 4, 4, 3, 4, 6, 3, 5, 2, 1, 8, 3, 10, 6, 5, 5, 9, 7, 6, 3, 8, 7, 9, 2, 5, 5, 2, 2, 9, 7, 3, 5, 8, 7, 6, 8, 7, 9, 9, 6, 3, 7, 8, 14, 5, 9, 10, 8, 11
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OFFSET

1,5


COMMENTS

Conjecture: (i) a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 3, 4, 8, 10, 24, 41.
(ii) For any integer n > 6, there is a positive integer k < n with n^2 + pi(k*n)  1 prime.
(iii) If n > 2, then pi(n^2)  pi(k*n) is prime for some 0 < k < n. If n > 1, then pi(n^2) + pi(k*n)  1 is prime for some 0 < k < n.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..4000
ZhiWei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 20142016.


EXAMPLE

a(2) = 1 since 2^2  pi(1*2) = 4  1 = 3 is prime.
a(3) = 1 since 3^2  pi(1*3) = 9  2 = 7 is prime.
a(4) = 1 since 4^2  pi(3*4) = 16  5 = 11 is prime.
a(8) = 1 since 8^2  pi(4*8) = 64  11 = 53 is prime.
a(10) = 1 since 10^2  pi(6*10) = 100  17 = 83 is prime.
a(24) = 1 since 24^2  pi(14*24) = 576  67 = 509 is prime.
a(41) = 1 since 41^2  pi(10*41) = 1681  80 = 1601 is prime.


MATHEMATICA

p[k_, n_]:=PrimeQ[n^2PrimePi[k*n]]
a[n_]:=Sum[If[p[k, n], 1, 0], {k, 1, n1}]
Table[a[n], {n, 1, 80}]


CROSSREFS

Cf. A000040, A000720, A237578, A237615, A237712, A238570.
Sequence in context: A022307 A029413 A237523 * A238421 A105154 A076447
Adjacent sequences: A238565 A238566 A238567 * A238569 A238570 A238571


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Feb 28 2014


STATUS

approved



