A237578 a(n) = |{0 < k < n: pi(k*n) is prime}|, where pi(.) is given by A000720.

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

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

Zhi-Wei Sun, Table of n, a(n) for n = 1..2500

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

Zhi-Wei 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, n-1}]
Table[a[n], {n, 1, 80}]

Crossrefs

Cf. A000040, A000720, A237453, A237496, A237497.

Sequence in context: A190683 A181810 A339304 * A026146 A325519 A221057

Adjacent sequences: A237575 A237576 A237577 * A237579 A237580 A237581

Keyword

nonn

Author

Zhi-Wei Sun, Feb 09 2014

Status

approved