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A262730
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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.
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3
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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
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OFFSET
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1,1
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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nonn,hard
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AUTHOR
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STATUS
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approved
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