Editing Carmichael number (section)

=== Factorizations ===
Carmichael numbers have at least three positive prime factors. The first Carmichael numbers with <math>k = 3, 4, 5, \ldots</math> prime factors are {{OEIS|id=A006931}}:

{| class="wikitable"
|-
! ''k'' !! &nbsp;
|-
| 3 || <math>561 = 3 \cdot 11 \cdot 17\,</math>
|-
| 4 || <math>41041 = 7 \cdot 11 \cdot 13 \cdot 41\,</math>
|-
| 5 || <math>825265 = 5 \cdot 7 \cdot 17 \cdot 19 \cdot 73\,</math>
|-
| 6 || <math>321197185 = 5 \cdot 19 \cdot 23 \cdot 29 \cdot 37 \cdot 137\,</math>
|-
| 7 || <math>5394826801 = 7 \cdot 13 \cdot 17 \cdot 23 \cdot 31 \cdot 67 \cdot 73\,</math>
|-
| 8 || <math>232250619601 = 7 \cdot 11 \cdot 13 \cdot 17 \cdot 31 \cdot 37 \cdot 73 \cdot 163\,</math>
|-
| 9 || <math>9746347772161 = 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 31 \cdot 37 \cdot 41 \cdot 641\,</math>
|}

The first Carmichael numbers with 4 prime factors are {{OEIS|id=A074379}}:

{| class="wikitable"
|-
! ''i'' !!&nbsp;
|-
| 1 || <math>41041 = 7 \cdot 11 \cdot 13 \cdot 41\,</math>
|-
| 2 || <math>62745 = 3 \cdot 5 \cdot 47 \cdot 89\,</math>
|-
| 3 || <math>63973 = 7 \cdot 13 \cdot 19 \cdot 37\,</math>
|-
| 4 || <math>75361 = 11 \cdot 13 \cdot 17 \cdot 31\,</math>
|-
| 5 || <math>101101 = 7 \cdot 11 \cdot 13 \cdot 101\,</math>
|-
| 6 || <math>126217 = 7 \cdot 13 \cdot 19 \cdot 73\,</math>
|-
| 7 || <math>172081 = 7 \cdot 13 \cdot 31 \cdot 61\,</math>
|-
| 8 || <math>188461 = 7 \cdot 13 \cdot 19 \cdot 109\,</math>
|-
| 9 || <math>278545 = 5 \cdot 17 \cdot 29 \cdot 113\,</math>
|-
| 10 || <math>340561 = 13 \cdot 17 \cdot 23 \cdot 67\,</math>
|}

The second Carmichael number (1105) can be expressed as the sum of two squares in more ways than any smaller number. The third Carmichael number (1729) is the [[1729 (number)|Hardy-Ramanujan Number]]: the smallest number that can be expressed as the [[sum of two cubes]] (of positive numbers) in two different ways.