OFFSET
1,1
COMMENTS
Conjecture: (i) The sequence has infinitely many terms. Also, there are infinitely many positive integers m such that pi(m^2) = pi(j^2)*pi(k^2) for no 0 < j <= k < m.
(ii) For any integer n > 2, the equation pi(x^n)*pi(y^n) = pi(z^n) has no solution with 0 < x <= y < z.
REFERENCES
Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..300
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.
EXAMPLE
a(1) = 8 since pi(8^2) = pi(64) = 18 = 2*9 = pi(2^2)*pi(5^2) with 0 < 2 < 5 < 8.
a(4) = 19 since pi(19^2) = pi(361) = 72 = 4*18 = pi(3^2)*pi(8^2) with 0 < 3 < 8 < 19.
MATHEMATICA
f[n_]:=PrimePi[n^2]
T[n_]:=Table[f[k], {k, 1, n}]
Dv[n_]:=Divisors[f[n]]
Le[n_]:=Length[Dv[n]]
n=0; Do[Do[If[MemberQ[T[m], Part[Dv[m], i]]&&MemberQ[T[m], Part[Dv[m], Le[m]-i+1]], n=n+1; Print[n, " ", m]; Goto[aa]], {i, 2, (Le[m]-1)/2}]; Label[aa]; Continue, {m, 1, 1080}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Sep 23 2015
STATUS
approved