

A262722


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


1



1, 41, 56, 74, 103, 157, 384, 491, 537, 868, 1490, 1710
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OFFSET

1,2


COMMENTS

Conjecture: (i) There are infinitely many distinct primes p,q,r such that pi(p^2+q^2) = r^2.
(ii) The Diophantine equation pi(x^3+y^3) = z^3 with 0 < x <= y and z > 0 only has the following 13 solutions: (x,y,z) = (1,1,1), (5,41,19), (47,56,29), (28,74,33), (2,103,44), (3,103,44), (6,157,65), (235,384,160), (266,491,198), (91,537,206), (359,868,331), (783,1490,565), (1192,1710,677).
(iii) The Diophantine equation pi(x^n+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 12 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..12.
ZhiWei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.


EXAMPLE

a(2) = 41 since pi(5^3+41^3) = pi(125+68921) = pi(69046) = 6859 = 19^3.


MATHEMATICA

f[x_, y_]:=PrimePi[x^3+y^3]
CQ[n_]:=IntegerQ[n^(1/3)]
n=0; Do[Do[If[CQ[f[x, y]], n=n+1; Print[n, " ", y]; Goto[aa]], {x, 1, y}]; Label[aa]; Continue, {y, 1, 1800}]


CROSSREFS

Cf. A000040, A000290, A000578, A000720, A262408, A262409, A262443, A262462, A262536, A262698, A262700, A262707.
Sequence in context: A118636 A116345 A127333 * A172406 A161613 A210338
Adjacent sequences: A262719 A262720 A262721 * A262723 A262724 A262725


KEYWORD

nonn,more,hard


AUTHOR

ZhiWei Sun, Sep 28 2015


STATUS

approved



