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A262462 Positive integers k with pi(k^3) a square, where pi(x) denotes the number of primes not exceeding x. 7

%I

%S 1,2,3,14,1122

%N Positive integers k with pi(k^3) a square, where pi(x) denotes the number of primes not exceeding x.

%C Conjecture: (i) The Diophantine equation pi(x^2) = y^2 with x > 0 and y > 0 has infinitely many solutions.

%C (ii) The only solutions to the Diophantine equation pi(x^m) = y^n with {m,n} = {2,3}, x > 0 and y > 0 are as follows:

%C pi(89^2) = 10^3, pi(2^3) = 2^2, pi(3^3) = 3^2, pi(14^3) = 20^2 and pi(1122^3) = 8401^2.

%C (iii) For m > 1 and n > 1 with m + n > 5, the equation pi(x^m) = y^n with x > 0 and y > 0 has no integral solution.

%C The conjecture seems reasonable in view of the heuristic arguments.

%C Part (ii) of the conjecture implies that the only terms of the current sequence are 1, 2, 3, 14 and 1122.

%e a(1) = 1 since pi(1^3) = 0^2.

%e a(2) = 2 since pi(2^3) = 2^2.

%e a(3) = 3 since pi(3^3) = 3^2.

%e a(4) = 14 since pi(14^3) = pi(2744) = 400 = 20^2.

%e a(5) = 1122 since pi(1122^3) = pi(1412467848) = 70576801 = 8401^2.

%t SQ[n_]:=IntegerQ[Sqrt[n]]

%t f[n_]:=PrimePi[n^3]

%t n=0;Do[If[SQ[f[k]],n=n+1;Print[n," ",k]],{k,1,1200}]

%Y Cf. A000290, A000578, A000720, A064523, A262408, A262409, A262443.

%K nonn,more

%O 1,2

%A _Zhi-Wei Sun_, Sep 23 2015

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Last modified June 1 03:14 EDT 2020. Contains 334758 sequences. (Running on oeis4.)