A225808 Values (Sum_{1<=i<=k} x_i)^2 = Sum_{1<=i<=k} x_i^3 for 1 <= x_1 <= x_2 <=...<= x_k ordered lexicographically according to (x1, x2,..., xk).

1, 9, 16, 36, 81, 81, 100, 144, 256, 169, 225, 324, 361, 625, 144, 256, 324, 441, 324, 361, 441, 625, 256, 576, 729, 784, 576, 729, 900, 961, 1089, 1296, 484, 625, 784, 900, 484, 441, 576, 729, 784, 900, 1089, 1089, 1156, 1369, 625, 784, 729, 900, 1089, 1369, 1296, 1600, 900, 961, 1089

Offset: 1

Author

Charles R Greathouse IV, Jimmy Zotos, and Balarka Sen, Jul 29 2013

Keywords

• nonn
• tabf

Comments

a(n) <= k^4 where k is the size of the ordered tuple (x_1, x_2,..., x_k).

This sequence is closed under multiplication, that is, if m and n are in this sequence, so is m*n.

Examples

			1;
9, 16;
36, 81;
81, 100, 144, 256;
169, 225, 324, 361, 625;
144, 256, 324, 441, 324, 361, 441, 625, 256, 576, 729, 784, 576, 729, 900, 961, 1089, 1296;
484, 625, 784, 900, 484, 441, 576, 729, 784, 900, 1089, 1089, 1156, 1369, 625, 784, 729, 900, 1089, 1369, 1296, 1600, 900, 961, 1089, 1600, 1296, 1600, 2025, 2401;
		

Links

• Balarka Sen, Rows n = 1..10 of irregular triangle, flattened
• Edward Barbeau and Samer Seraj, Sum of cubes is square of sum, arXiv:1306.5257 [math.NT], 2013.
• John Mason, Generalising 'sums of cubes equal to squares of sums', The Mathematical Gazette 85:502 (2001), pp. 50-58.
• Alasdair McAndrew, A cute result relating to sums of cubes
• David Pagni, 82.27 An interesting number fact, The Mathematical Gazette 82:494 (1998), pp. 271-273.
• Balarka Sen, Table of rows, n = 1..10
• W. R. Utz, The Diophantine Equation (x_1 + x_2 + ... + x_n)^2 = x_1^3 + x_2^3 + ... + x_n^3, Fibonacci Quarterly 15:1 (1977), pp. 14, 16. Part 1, part 2.

Crossrefs

Cf. A158649, A055012, A118881.

Programs

Mathematica

row[n_] := Reap[Module[{v, m}, v = Table[1, {n}]; m = n^(4/3); While[ v[[-1]] < m, v[[1]]++; If[v[[1]] > m, For[i = 2, i <= m, i++, If[v[[i]] < m, v[[i]]++; For[j = 1, j <= i - 1, j++, v[[j]] = v[[i]]]; Break[]]]]; If[Total[v^3] == Total[v]^2, Sow[Total[v]^2]]]]][[2, 1]];
Array[row, 7] // Flatten (* Jean-François Alcover, Feb 23 2019, from PARI *)

PARI

row(n)=my(v=vector(n,i,1),N=n^(4/3)); while(v[#v]N,for(i=2, N,if(v[i]