A377372 a(n) is the smallest prime p such that the Diophantine equation x^3 + y^3 + z^3 = p^3, where 0 < x <= y <= z has exactly n positive integer solutions.
2, 19, 41, 479, 1031, 1181, 577, 2999, 10711, 29033, 24919, 49069, 60919, 169019, 209563, 254993, 337537
Offset: 0
Examples
a(3)=479, because 479^3 = 47^3 + 350^3 + 406^3 = 109^3 + 293^3 + 437^3 = 256^3 + 311^3 + 398^3 and no prime less than 479 has 3 solutions.
Programs
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Mathematica
a = Table[SelectFirst[Table[{p,Length@Select[PowersRepresentations[p^3, 3, 3], #[[1]] > 0 &]}, {p, Prime@Range@200}], #[[2]] == k &], {k, 0, 6}]
Extensions
a(11) from Jinyuan Wang, May 31 2025
a(12) from Chai Wah Wu, Jun 03 2025
a(13)-a(15) from Chai Wah Wu, Jun 04 2025
a(16) from Chai Wah Wu, Jun 10 2025