A262730 Primes p such that p^2 = pi(x^3) + pi(y^3) for some positive integers x and y, where pi(m) denotes the number of primes not exceeding m.

2, 3, 23, 83, 199, 331, 487, 1069, 1289, 1697, 2467, 3463, 3617, 3733, 5153, 5449, 6221, 9203, 9811, 9967, 12473, 13883, 14723, 15791, 16001, 18919, 33589, 33827, 46093, 58321, 59051, 59921, 60289, 71249, 84349, 85133, 88211, 124309, 126047, 126359, 127541, 145679, 146807, 153247, 165233

Offset: 1

Author

Zhi-Wei Sun, Sep 28 2015

Keywords

• nonn
• hard

Comments

Conjecture: (i) The sequence has infinitely many terms.

(ii) There are infinitely many primes p such that p^2 = pi(x^3+y^3) for some positive integers x and y.

See also A262731 for a related conjecture.

Examples

			a(1) = 2 since pi(1^3)+pi(2^3) = 0+4 = 2^2 with 2 prime.
a(3) = 23 since pi(9^3)+pi(14^3) = pi(729)+pi(2744) = 129+400 = 529 = 23^2 with 23 prime.
a(20) = 9967 since pi(841^3)+pi(1109^3) = pi(594823321)+pi(1363938029) = 31068537+68272552 = 99341089 = 9967^2 with 9967 prime.
a(38) = 124309 since pi(5773^3)+pi(5779^3) = pi(192399824917)+pi(193000344139) = 7714808769+7737918712 = 15452727481 = 124309^2 with 124309 prime.
a(45) = 165233 since pi(6924^3)+pi(7148^3) = pi(331948857024)+pi(365219225792) = 13025048890+14276895399 = 27301944289 = 165233^2 with 165233 prime.
		

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

• Chai Wah Wu, Table of n, a(n) for n = 1..132
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

Cf. A000040, A000290, A000578, A000720, A262409, A262462, A262536, A262698, A262722, A262731.

Programs

Mathematica

f[n_]:=PrimePi[n^3]
T[1]:={0}
T[n_]:=Union[T[n-1],{f[n]}]
n=0;Do[Do[If[f[x]>Prime[y]^2,Goto[aa]];If[MemberQ[T[Prime[y]],Prime[y]^2-f[x]],n=n+1;Print[n," ",Prime[y]];Goto[aa]];Continue,{x,1,Prime[y]}];
Label[aa];Continue,{y,1,15111}]