A385566 -id:A385566 - cp's OEIS frontend

A385565 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 = k^4, where 0 < x < y < z has exactly n integer solutions.

11, 21, 64, 144, 330, 846, 342, 252, 1331, 1008, 720, 1890, 3780, 729, 4200, 2016, 1000, 216, 6300, 8352, 10800, 12312, 8568, 19440, 8280, 9576, 21204

Offset: 1

Zhining Yang, Jul 03 2025

a(13) and a(15) not found up to k = 3300, a(14) = 729, a(16) = 2016, a(17) = 1000, a(18) = 216.

			a(3)=64, because 64^4 = 9^3 + 58^3 + 255^3 = 9^3 + 183^3 + 220^3 = 22^3 + 57^3 + 255^3 and no integer less than 64 has 3 solutions.

Cf. A024975, A025419, A377444, A385354, A385566.

s = Table[{k, Length@Select[PowersRepresentations[k^4, 3, 3],
      0 < #[[1]] < #[[2]] < #[[3]] &]}, {k, 400}];
a = Table[SelectFirst[s, #[[2]] == k &], {k, 5}][[All, 1]]

a(13), a(15), a(19)-a(21) from Chai Wah Wu, Jul 08 2025

a(22)-a(27) from Chai Wah Wu, Jul 18 2025

A385714 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 = k^n, where 0 < x < y < z has an integer solution, or -1 if no such integer exists.

Original entry on oeis.org

36, 6, 6, 11, 6, 3, 6, 6, 3, 6, 6, 3, 6, 6, 3, 6, 6, 3, 6, 6, 3, 6, 3, 2, 6, 3, 2, 6, 3, 2, 6, 3, 2, 6, 3, 2

Offset: 1

Jean-Marc Rebert, Jul 07 2025

			n [x, y, z]         k^n
1 [1, 2, 3]       36 = 36^1;
2 [1, 2, 3]       36 = 6^2;
3 [3, 4, 5]      216 = 6^3;
4 [12, 17, 20] 14641 = 11^4;
5 [6, 12, 18]   7776 = 6^5;
6 [1, 6, 8]      729 = 3^6;

Cf. A385354, A385565, A384430, A385566.

For 4 < n <= 22, if n == 0 (mod 3) then a(n) = 3, else a(n)=6.
From Chai Wah Wu, Jul 15 2025: (Start)
Conjecture: if n >= 24 and n == 0 (mod 3), then a(n) = 2. Verified for n <= 69.
Conjecture: if n >= 23 and n == 2 (mod 3), then a(n) = 3. Verified for n <= 44.
Conjecture: if n >= 7 and n == 1 (mod 3), then a(n) = 6. (End)
From David A. Corneth, Jul 15 2025: (Start)
For n >= 24 and n == 0 (mod 3) indeed we have a(n) = 2. Proof: holds for n = 24. For multiples of 3 that are >= 27 and m >= 0 we have:
2^(27 + 3*m) = (18 * 2^m)^3 + (366 * 2^m)^3 + (440 * 2^m)^3.
In general 2 <= a(n+3) <= a(n) therefore a(n) <= 6 for n >= 4. (End).

a(7)-a(12) from David A. Corneth, Jul 07 2025
a(16)-a(27) from Chai Wah Wu, Jul 15 2025
a(28)-a(34) from David A. Corneth, Jul 15 2025, Jul 17 2025

cp's OEIS Frontend

A385565 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 = k^4, where 0 < x < y < z has exactly n integer solutions.

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