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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).

%I A225808 #38 Jan 05 2025 19:51:40
%S A225808 1,9,16,36,81,81,100,144,256,169,225,324,361,625,144,256,324,441,324,
%T A225808 361,441,625,256,576,729,784,576,729,900,961,1089,1296,484,625,784,
%U A225808 900,484,441,576,729,784,900,1089,1089,1156,1369,625,784,729,900,1089,1369,1296,1600,900,961,1089
%N 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).
%C A225808 a(n) <= k^4 where k is the size of the ordered tuple (x_1, x_2,..., x_k).
%C A225808 This sequence is closed under multiplication, that is, if m and n are in this sequence, so is m*n.
%H A225808 Balarka Sen, <a href="/A225808/b225808.txt">Rows n = 1..10 of irregular triangle, flattened</a>
%H A225808 Edward Barbeau and Samer Seraj, <a href="http://arxiv.org/abs/1306.5257">Sum of cubes is square of sum</a>, arXiv:1306.5257 [math.NT], 2013.
%H A225808 John Mason, <a href="http://www.jstor.org/stable/3620469">Generalising 'sums of cubes equal to squares of sums'</a>, The Mathematical Gazette 85:502 (2001), pp. 50-58.
%H A225808 Alasdair McAndrew, <a href="http://amca01.wordpress.com/2011/01/10/a-cute-result-relating-to-sums-of-cubes/">A cute result relating to sums of cubes</a>
%H A225808 David Pagni, <a href="http://www.jstor.org/stable/3620410">82.27 An interesting number fact</a>, The Mathematical Gazette 82:494 (1998), pp. 271-273.
%H A225808 Balarka Sen, <a href="/A225808/a225808.txt">Table of rows, n = 1..10</a>
%H A225808 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. <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/15-1/utz-a.pdf">Part 1</a>, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/15-1/utz-b.pdf">part 2</a>.
%e A225808 1;
%e A225808 9, 16;
%e A225808 36, 81;
%e A225808 81, 100, 144, 256;
%e A225808 169, 225, 324, 361, 625;
%e A225808 144, 256, 324, 441, 324, 361, 441, 625, 256, 576, 729, 784, 576, 729, 900, 961, 1089, 1296;
%e A225808 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;
%t A225808 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]];
%t A225808 Array[row, 7] // Flatten (* _Jean-François Alcover_, Feb 23 2019, from PARI *)
%o A225808 (PARI) row(n)=my(v=vector(n,i,1),N=n^(4/3)); while(v[#v]<N, v[1]++; if(v[1]>N,for(i=2, N,if(v[i]<N,v[i]++;for(j=1,i-1, v[j]=v[i]); break))); if(sum(i=1,n,v[i]^3)==sum(i=1,n,v[i])^2,print1(sum(i=1,n,v[i])^2", ")))
%Y A225808 Cf. A158649, A055012, A118881.
%K A225808 nonn,tabf
%O A225808 1,2
%A A225808 _Charles R Greathouse IV_, _Jimmy Zotos_, and _Balarka Sen_, Jul 29 2013