

A237578


a(n) = {0 < k < n: pi(k*n) is prime}, where pi(.) is given by A000720.


31



0, 0, 2, 2, 1, 3, 2, 1, 2, 2, 4, 4, 1, 4, 2, 5, 5, 6, 2, 5, 4, 6, 3, 7, 3, 3, 7, 5, 5, 5, 10, 9, 3, 7, 6, 5, 12, 3, 3, 9, 10, 11, 12, 7, 3, 5, 11, 9, 7, 10, 12, 9, 10, 8, 12, 11, 10, 17, 15, 13, 14, 18, 4, 17, 10, 9, 15, 11, 14, 11, 23, 11, 9, 13, 12, 12, 12, 11, 14, 16
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OFFSET

1,3


COMMENTS

Conjecture: a(n) > 0 for all n > 2, and a(n) = 1 only for n = 5, 8, 13. Moreover, for each n = 1, 2, 3, ..., there is a positive integer k < 3*sqrt(n) + 3 with pi(k*n) prime.
Note that the least positive integer k with pi(k*38) prime is 21 < 3*sqrt(38) + 3 < 21.5.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..2500
ZhiWei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 20142016.
ZhiWei Sun and Lilu Zhao, On the set {pi(kn): k=1,2,3,...}, arXiv:2004.01080 [math.NT], 2020.


EXAMPLE

a(5) = 1 since pi(1*5) = 3 is prime.
a(8) = 1 since pi(4*8) = 11 is prime.
a(13) = 1 since pi(10*13) = pi(130) = 31 is prime.
a(38) = 3 since pi(21*38) = pi(798) = 139, pi(28*38) = pi(1064) = 179 and pi(31*38) = pi(1178) = 193 are all prime.


MATHEMATICA

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


CROSSREFS

Cf. A000040, A000720, A237453, A237496, A237497.
Sequence in context: A281013 A190683 A181810 * A026146 A325519 A221057
Adjacent sequences: A237575 A237576 A237577 * A237579 A237580 A237581


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Feb 09 2014


STATUS

approved



