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

1,4

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.

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

Example

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.

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}]

Crossrefs

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

Sequence in context: A239858 A031437 A282561 * A138241 A234615 A029145

Adjacent sequences: A237595 A237596 A237597 * A237599 A237600 A237601

Keyword

nonn

Author

Zhi-Wei Sun, Feb 10 2014

Status

approved