A225808 - OEIS

%I #38 Jan 05 2025 19:51:40

%S 1,9,16,36,81,81,100,144,256,169,225,324,361,625,144,256,324,441,324,

%T 361,441,625,256,576,729,784,576,729,900,961,1089,1296,484,625,784,

%U 900,484,441,576,729,784,900,1089,1089,1156,1369,625,784,729,900,1089,1369,1296,1600,900,961,1089

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

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

%H Balarka Sen, <a href="/A225808/b225808.txt">Rows n = 1..10 of irregular triangle, flattened</a>

%H 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 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 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 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 Balarka Sen, <a href="/A225808/a225808.txt">Table of rows, n = 1..10</a>

%H 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 1;

%e 9, 16;

%e 36, 81;

%e 81, 100, 144, 256;

%e 169, 225, 324, 361, 625;

%e 144, 256, 324, 441, 324, 361, 441, 625, 256, 576, 729, 784, 576, 729, 900, 961, 1089, 1296;

%e 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 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 Array[row, 7] // Flatten (* _Jean-François Alcover_, Feb 23 2019, from PARI *)

%o (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 Cf. A158649, A055012, A118881.

%K nonn,tabf

%O 1,2

%A _Charles R Greathouse IV_, _Jimmy Zotos_, and _Balarka Sen_, Jul 29 2013