

A262698


Positive integers m such that pi(k^3)+pi(m^3) is a cube for some k = 1,...,m, where pi(x) denotes the number of primes not exceeding x.


6



1, 2, 4, 24, 41, 51, 88, 95, 99, 179, 183, 663, 782, 829, 1339, 2054, 2816
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OFFSET

1,2


COMMENTS

Conjecture: (i) There are infinitely many distinct primes p,q,r such that pi(p^2) + pi(q^2) = r^2.
(ii) The Diophantine equation pi(x^3) + pi(y^3) = z^3 with 0 < x <= y and z >= 0 only has the following 17 solutions: (x,y,z) = (1,1,0), (2,2,2), (3,4,3), (16,24,13),(3,41,19), (37,51,26), (53,88,41), (18,95,41), (45,99,44),(108,179,79), (149,183,87), (8,663,251), (243,782,297),(803,829,385), (100,1339,489), (674,2054,745),(1519,2816,1047).
(iii) The Diophantine equation pi(x^n) + pi(y^n) = z^n with n > 3 and x,y,z > 0 has no solution.
Part (ii) of the conjecture implies that the current sequence only has 17 terms as shown here.


REFERENCES

ZhiWei 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 ChinaJapan Seminar (Fukuoka, Oct. 28  Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169187.


LINKS

Table of n, a(n) for n=1..17.
ZhiWei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.


EXAMPLE

a(4) = 24 since pi(16^3) + pi(24^3) = pi(4096) + pi(13824) = 564 + 1633 = 2197 = 13^3.


MATHEMATICA

f[n_]:=PrimePi[n^3]
CQ[n_]:=IntegerQ[n^(1/3)]
n=0; Do[Do[If[CQ[f[x]+f[y]], n=n+1; Print[n, " ", y]; Goto[aa]], {x, 1, y}]; Label[aa]; Continue, {y, 1, 3000}]


CROSSREFS

Cf. A000578, A000720, A019590, A262408, A262409, A262447, A262462, A262536.
Sequence in context: A171459 A240558 A163896 * A168054 A280075 A068506
Adjacent sequences: A262695 A262696 A262697 * A262699 A262700 A262701


KEYWORD

nonn,more


AUTHOR

ZhiWei Sun, Sep 27 2015


STATUS

approved



