A237598 a(n) = |{0 < k < prime(n): pi(k*n) is a square}|, where pi(.) is given by A000720.

1, 1, 1, 2, 2, 2, 4, 3, 5, 2, 3, 5, 3, 6, 1, 2, 3, 3, 5, 3, 5, 2, 6, 4, 4, 5, 3, 6, 4, 3, 2, 5, 3, 4, 3, 4, 4, 3, 6, 4, 3, 4, 2, 1, 2, 9, 3, 4, 4, 4, 5, 7, 4, 7, 3, 6, 7, 3, 7, 7, 5, 1, 4, 5, 3, 3, 10, 5, 4, 7

Offset: 1

Author

Zhi-Wei Sun, Feb 10 2014

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 0.

(ii) For each n > 9, there is a positive integer k < prime(n)/2 such that pi(k*n) is a triangular number.

See also A237612 for the least k > 0 with pi(k*n) a square.

Examples

			a(3) = 1 since pi(3*3) = 2^2 with 3 < prime(3) = 5.
a(6) = 2 since pi(4*6) = 3^2 with 4 < prime(6) = 13, and pi(9*6) =  4^2 with 9 < prime(6) = 13.
a(15) = 1 since pi(28*15) = 9^2 with 28 < prime(15) = 47.
a(62) = 1 since pi(68*62) = 24^2 with 68 < prime(62) = 293.
a(459) = 1 since pi(2544*459) = 301^2 with 2544 < prime(459) = 3253.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..2500
• Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Crossrefs

Cf. A000040, A000217, A000290, A000720, A237578, A237597, A237612, A237614.

Programs

Mathematica

sq[n_]:=IntegerQ[Sqrt[PrimePi[n]]]
a[n_]:=Sum[If[sq[k*n],1,0],{k,1,Prime[n]-1}]
Table[a[n],{n,1,70}]