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

Author

Zhi-Wei Sun, Feb 09 2014

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

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

Programs

Mathematica

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