A387023 Integers w such that the Diophantine equation x^2 + y^3 + z^4 = w^5 where GCD(x,y,z)=1 has exactly 5 positive integer solutions.

9, 45, 70, 80, 120, 124, 125, 128, 133, 143, 170, 175, 180, 195, 201, 220, 224, 236, 252, 264, 275, 278, 296, 308, 311, 312, 330, 332, 336, 337, 352, 354, 355, 360, 362, 366, 374, 375, 380, 386, 390, 394, 399, 404, 411, 416, 418, 428, 430, 444, 461, 466, 477, 484, 488, 500

Offset

1,1

Links

Zhining Yang, Table of n, a(n) for n = 1..328

Example

444 is in the sequence because 444^5 = x^2 + y^3 + z^4 where GCD (x, y, z) = 1 has exactly 5 positive integer solutions: {676786, 25603, 343}, {342332, 25775, 345}, {4123199, 5503, 544}, {2451712, 21919, 919}, {3889117, 679, 1208}.

Mathematica

Do[w5=w^5; s={}; c=0;
Do[yy=w5-z^4; Do[xx=yy-y^3; x=Sqrt@xx;
If[IntegerQ@x, If[GCD[x, y, z]==1, c++; AppendTo[s, {x, y, z}]]], {y, Floor[yy^(1/3)]}], {z, Floor[w5^(1/4)]}];
If[c==5, Print[w, s]], {w, 100}]

Crossrefs

Cf. A386373, A386377, A386521.

Sequence in context: A044492 A207359 A243090 * A067536 A139609 A321779

Adjacent sequences: A387020 A387021 A387022 * A387024 A387025 A387026

Keyword

nonn

Author

Zhining Yang, Aug 13 2025

Status

approved