A262443 Positive integers m such that pi(m^2) = pi(j^2)*pi(k^2) for some 0 < j < k < m, where pi(x) denotes the number of primes not exceeding x.

8, 11, 14, 19, 20, 36, 38, 45, 66, 87, 91, 115, 139, 143, 152, 155, 201, 220, 227, 279, 357, 383, 391, 415, 418, 452, 476, 480, 489, 496, 500, 514, 521, 524, 549, 552, 557, 588, 595, 632, 653, 676, 706, 708, 749, 753, 761, 766, 820, 846, 863, 877, 922, 1009, 1038, 1041, 1044, 1052, 1057, 1080

Offset: 1

Author

Zhi-Wei Sun, Sep 23 2015

Keywords

• nonn

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.

Examples

			 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.
		

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.

Crossrefs

Cf. A000290, A000720, A038107, A262408, A262409.

Programs

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}]