A biquadratic number is a fourth power, 
. The first few biquadratic numbers are 1, 16, 81, 256, 625,
... (OEIS A000583). The minimum number of biquadratic
numbers needed to represent the numbers 1, 2, 3, ... are 1, 2, 3, 4, 5, 6, 7, 8,
9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, ... (OEIS A002377),
and the number of distinct ways to represent the numbers 1, 2, 3, ... in terms of
biquadratic numbers are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2,
2, .... A brute-force algorithm for enumerating the biquadratic permutations of 
is repeated application of the greedy
algorithm.
Every positive integer is expressible as a sum of (at most) 
biquadratic numbers (Waring's
problem). Davenport (1939) showed that 
, meaning that all sufficiently large integers require
only 16 biquadratic numbers. It is also known that every integer is a sum of at most
10 signed biquadrates (
; although it is not known if 10 can be reduced
to 9). The following table gives the first few numbers which require 1, 2, 3, ...,
19 biquadratic numbers to represent them as a sum, with the sequences for 17, 18,
and 19 being finite.
| # | OEIS | numbers |
| 1 | A000583 | 1, 16, 81, 256, 625, 1296, 2401, 4096, ... |
| 2 | A003336 | 2, 17, 32, 82, 97, 162, 257, 272, ... |
| 3 | A003337 | 3, 18, 33, 48, 83, 98, 113, 163, ... |
| 4 | A003338 | 4, 19, 34, 49, 64, 84, 99, 114, 129, ... |
| 5 | A003339 | 5, 20, 35, 50, 65, 80, 85, 100, 115, ... |
| 6 | A003340 | 6, 21, 36, 51, 66, 86, 96, 101, 116, ... |
| 7 | A003341 | 7, 22, 37, 52, 67, 87, 102, 112, 117, ... |
| 8 | A003342 | 8, 23, 38, 53, 68, 88, 103, 118, 128, ... |
| 9 | A003343 | 9, 24, 39, 54, 69, 89, 104, 119, 134, ... |
| 10 | A003344 | 10, 25, 40, 55, 70, 90, 105, 120, 135, ... |
| 11 | A003345 | 11, 26, 41, 56, 71, 91, 106, 121, 136, ... |
| 12 | A003346 | 12, 27, 42, 57, 72, 92, 107, 122, 137, ... |
| 13 | A046044 | 13, 28, 43, 58, 73, 93, 108, 123, 138, ... |
| 14 | A046045 | 14, 29, 44, 59, 74, 94, 109, 124, 139, ... |
| 15 | A046046 | 15, 30, 45, 60, 75, 95, 110, 125, 140, ... |
| 16 | A046047 | 31, 46, 61, 76, 111, 126, 141, 156, ... |
| 17 | A046048 | 47, 62, 77, 127, 142, 157, 207, 222, ... |
| 18 | A046049 | 63, 78, 143, 158, 223, 238, 303, 318, ... |
| 19 | A046050 | 79, 159, 239, 319, 399 |
The following table gives the numbers which can be represented in 
different ways as a sum of 
biquadrates.
 |  | OEIS | numbers |
| 1 | 1 | A000583 | 1, 16, 81, 256, 625, 1296, 2401, 4096, ... |
| 2 | 2 | A018786 | 635318657, 3262811042, 8657437697, ... |
The numbers 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, ... (OEIS A046039) cannot be represented using distinct biquadrates.
