A004000

A004000 is a sequence in the On-Line Encyclopedia of Integer Sequences (OEIS). It is defined as the sequence of integers n such that n divides the sum of the cubes of the integers from 1 to n. In other words, for a given integer n, the condition for it to be in the sequence A004000 is that (1^3 + 2^3 + ... + n^3) mod n = 0.

The sum of the first n cubes is given by the formula (n(n+1)/2)^2. Therefore, the condition for n to be in A004000 can be rewritten as (n(n+1)/2)^2 mod n = 0. Expanding this, we get (n^2 * (n+1)^2 / 4) mod n = 0. For this to hold, the term n^2 * (n+1)^2 / 4 must be an integer multiple of n. This means n must divide (n^2 * (n+1)^2 / 4).

The first few terms of the sequence A004000 are 1, 2, 3, 4, 6, 8, 12, 16, 18, 24, 36, 48, 54, 64, 72, 96, 108, 128, 144, 162, 192, 216, 243, 256, 288, 324, 384, 432, 486, 512, 576, 648, 675, 729, 768, 864, 972, 1024, 1089, 1152, 1296, 1350, 1458, 1536, 1728, 1944, 2048, 2187, 2304, 2430, 2592, 2700, 2916, 3072, 3240, 3456, 3645, 3888, 4096.

The sequence has been studied in number theory, and it is related to properties of divisibility. It is known that all powers of 2 are in the sequence.