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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 (list; graph; refs; listen; history; text; internal format)
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

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

Table of n, a(n) for n=1..17.

Zhi-Wei 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

Zhi-Wei Sun, Sep 27 2015

STATUS

approved

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Last modified August 20 08:50 EDT 2018. Contains 313914 sequences. (Running on oeis4.)