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

1, 2, 3, 14, 1122

Offset

1,2

Comments

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

(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:

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.

(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.

The conjecture seems reasonable in view of the heuristic arguments.

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

Links

Table of n, a(n) for n=1..5.

Example

a(1) = 1 since pi(1^3) = 0^2.
a(2) = 2 since pi(2^3) = 2^2.
a(3) = 3 since pi(3^3) = 3^2.
a(4) = 14 since pi(14^3) = pi(2744) = 400 = 20^2.
a(5) = 1122 since pi(1122^3) = pi(1412467848) = 70576801 = 8401^2.

Mathematica

SQ[n_]:=IntegerQ[Sqrt[n]]
f[n_]:=PrimePi[n^3]
n=0; Do[If[SQ[f[k]], n=n+1; Print[n, " ", k]], {k, 1, 1200}]

Crossrefs

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

Sequence in context: A042817 A224848 A271330 * A180698 A266618 A082572

Adjacent sequences: A262459 A262460 A262461 * A262463 A262464 A262465

Keyword

nonn,more

Author

Zhi-Wei Sun, Sep 23 2015

Status

approved