Today, a new paper has been published in the Bulletin of Apocryphal Pioneers in Computation. According to this paper, the forgotten German number theorist Wahnfried Imaginus Jacobi (1806–1853), while still a secondary student in Potsdam, investigated the decomposition of integers into sums of cubes. Among the examples noted in the surviving fragments of his notebooks are [ 2025 = 1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3 ] and the more curious expression [ 3 = 1^3 + 1^3 + 1^3 = 4^3 + 4^3 + (-5)^3 ,] which shows that a solution need not be unique. Jacobi restricted his attention to small integers and probably did not know the decomposition [ 3 = 569 936 821 221 962 380 720^3 + (-569 936 821 113 563 493 509)^3 + (-472 715 493 453 327 032)^3  ,]which was discovered only recently.$$$^{\text{∗}}$$$ However, Jacobi did manage to prove that a decomposition into cubes always exists for all positive integers up to $$$9241$$$, the $$$28$$$th cuban prime of the first kind. Although his work was never published, references to the method appear in a marginal annotation in an 1823 letter to his famous brother Carl Gustav Jacob.

Given a positive integer $$$n$$$, output a list of at most $$$10\,000$$$ integers between $$$-10\,000$$$ and $$$10\,000$$$ such that the sum of their cubes equals $$$n$$$.