A262722 - OEIS

%I #32 Apr 26 2021 20:54:14

%S 1,41,56,74,103,157,384,491,537,868,1490,1710,4322,4523,4877,4942,

%T 5147,5407,7564,17576,67722,131455,220641,438895,443475,553878,571473,

%U 625611

%N 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.

%C Conjecture: (i) There are infinitely many distinct primes p,q,r such that pi(p^2+q^2) = r^2.

%C (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).

%C (iii) The Diophantine equation pi(x^n+y^n) = z^n with n > 3 and x,y,z > 0 has no solution.

%C Part (ii) of the conjecture implies that the current sequence only has 12 terms as shown here.

%C Conjecture (ii) is false as there are more terms beyond 1710. It is likely the sequence has an infinite number of terms. (x,y,z) for 1710 < y <= 7564: (1429, 4322, 1514), (1974, 4523, 1604), (3361, 4877, 1840), (3992, 4942, 1949), (3253, 5147, 1902), (971, 5407, 1859), (935, 7564, 2563). - _Chai Wah Wu_, Apr 12 2021

%C Solutions (x,y,z) for 7564 < y <= 67722: (3484, 17576, 5783), (25184, 67722, 21604). - _Chai Wah Wu_, Apr 17 2021

%C Solutions (x,y,z) for 67722 < y <= 625611: (61021, 131455, 41715), (93577, 220641, 68507), (394510, 438895, 155930), (3086, 443475, 131933), (338485, 553878, 175133), (239982, 571473, 172855), (610794, 625611, 228409). - _Chai Wah Wu_, Apr 26 2021

%D 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.

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014.

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

%t f[x_,y_]:=PrimePi[x^3+y^3]

%t CQ[n_]:=IntegerQ[n^(1/3)]

%t 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}]

%Y Cf. A000040, A000290, A000578, A000720, A262408, A262409, A262443, A262462, A262536, A262698, A262700, A262707.

%K nonn,more,hard

%O 1,2

%A _Zhi-Wei Sun_, Sep 28 2015

%E a(13)-a(19) from _Chai Wah Wu_, Apr 12 2021

%E a(20)-a(21) from _Chai Wah Wu_, Apr 17 2021

%E a(22)-a(28) from _Chai Wah Wu_, Apr 26 2021