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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 (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+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

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..12.

Zhi-Wei 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

Zhi-Wei Sun, Sep 28 2015

STATUS

approved

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Last modified October 1 16:22 EDT 2020. Contains 337443 sequences. (Running on oeis4.)