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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). 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

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.

EXAMPLE

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;

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-Fran├žois Alcover, Feb 23 2019, from PARI *)

PROG

(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", ")))

CROSSREFS

Cf. A158649, A055012, A118881.

Sequence in context: A039788 A175652 A110833 * A076621 A134937 A076431

Adjacent sequences:  A225805 A225806 A225807 * A225809 A225810 A225811

KEYWORD

nonn,tabf

AUTHOR

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

STATUS

approved

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Last modified August 9 22:52 EDT 2020. Contains 336335 sequences. (Running on oeis4.)