A158649 Number of integral solutions to the equation (x_1)^3 + ... + (x_n)^3 = (x_1 + ... + x_n)^2 with 1 <= x_1 <= ... <= x_n.

1, 1, 2, 2, 4, 5, 18, 30, 94, 226, 715, 2024, 6546, 20622, 69459, 232406, 810943, 2828246

Offset

0,3

Comments

One prominent solution is x_i = i; another obvious one is x_i = n.

It is easy to show that in every solution (x_1, ..., x_n), the sum x_1 + ... + x_n <= n^2 and x_n <= n^(4/3).

There is only one solution with pairwise distinct x_i, it has x_i = i for all i. - Max Alekseyev, Sep 07 2010

x_1 + ... + x_n != 3k + 2. - David A. Corneth, Nov 06 2018

References

Titu Andreescu and Dorin Andrica, An Introduction To Diophantine Equations, 2002, GIL Publishing House, pp. 38, example 5.

Peter Giblin, Primes and Programming, 1993, Cambridge University Press. See chapter 9, exercise 1.7.

Links

Table of n, a(n) for n=0..17.

Max A. Alekseyev, Problem 3766, Crux Mathematicorum 38(7) (2012), 284-287.

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 (2011)

David Pagni, 82.27 An interesting number fact, The Mathematical Gazette 82:494 (1998), pp. 271-273.

C. Rivera, Puzzle 158. Sum of Cubes equal to Square of Sum (2001)

Greg Ross, Hocus Pocus

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.

Formula

A001055(n) <= a(n) << e^n n^(n/3). - Charles R Greathouse IV, May 24 2013

Example

a(4) = 4, since there are four solutions of length n=4: (1,2,2,4), (1,2,3,4), (2,2,4,4), and (4,4,4,4).

Mathematica

a[0] = a[1] = 1;
a[n_] := Module[{x}, cnt = 0; xx = Array[x, n]; m = Floor[n^(4/3)]; x[0] = 1; iter = Table[{x[k], x[k-1], m}, {k, 1, n}]; Do[If[Total[xx] <= n^2, If[Total[xx^3] == Total[xx]^2, cnt++]], Sequence @@ iter // Evaluate]; cnt];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 10}] (* Jean-François Alcover, Nov 06 2018 *)

Crossrefs

Cf. A227847.

Sequence in context: A302400 A175954 A371162 * A357886 A019964 A087377

Adjacent sequences: A158646 A158647 A158648 * A158650 A158651 A158652

Keyword

more,nonn

Author

Jens Voß, Mar 23 2009

Extensions

Edited by Max Alekseyev, Aug 18 2010

a(12)-a(13) from Max Alekseyev, Aug 20 2010

a(14) from Max Alekseyev, Sep 07 2010

a(15)-a(17) from Charles R Greathouse IV, Jun 05 2013

Status

approved