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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 10 2014
STATUS
approved