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.

1, 2, 4, 24, 41, 51, 88, 95, 99, 179, 183, 663, 782, 829, 1339, 2054, 2816, 7918, 8474, 13264, 16664, 27415, 39514, 48606, 51145, 187222, 200906, 261980, 353209, 375162, 396967, 400469

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.

a(26) > 10^5, if it exists. Conjecture (ii) above is false since these further solutions exist: (1339, 7918, 2682), (3360, 8474, 2922), (8443, 13264, 4764), (15590, 16664, 6696), (15883, 27415, 9431), (9719, 39514, 12689), (22265, 48606, 15933), (38606, 51145, 18297). - Giovanni Resta, Jun 14 2020

Further solutions: (79522, 187222, 58554), (65281, 200906, 61833), (222863, 261980, 92917), (226465, 353209, 114585), (41559, 375162, 112168), (244967, 396967, 127399), (291034, 400469, 133443) - Chai Wah Wu, Apr 13 2021

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..32.

Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2016.

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}]

Prog

(PARI) lista(nn) = {my(c=0, v=vector(nn)); for(m=1, nn, forprime(p=(m-1)^3+1, m^3, c++); v[m]=c; if(sum(k=1, m, ispower(v[k]+v[m], 3)), print1(m, ", "))); } \\ Jinyuan Wang, Jun 13 2020

Crossrefs

Cf. A000578, A000720, A019590, A262408, A262409, A262447, A262462, A262536.

Sequence in context: A240558 A163896 A350813 * A347946 A168054 A280075

Adjacent sequences: A262695 A262696 A262697 * A262699 A262700 A262701

Keyword

nonn,more

Author

Zhi-Wei Sun, Sep 27 2015

Extensions

a(18)-a(25) from Giovanni Resta, Jun 14 2020

a(26)-a(29) from Chai Wah Wu, Apr 05 2021

a(30)-a(32) from Chai Wah Wu, Apr 09 2021

Status

approved