A square-free number is one that does not contain a square as a factor. One interesting sequence involves non-square-free numbers: that is, numbers that are divisible by a square.
The first number in this sequence is 4, which is the smallest integer containing a non-trivial (i.e. not 0 or 1) square. The second number is 8, which is the first of a pair of consecutive integers that has a square factor (8 is divisible by 4, a square, and 9 is of course the square of 3).
The third number in the sequence is 48, which is the start of three consecutive integers divisible by a square: 48 is divisible by a square (4), 49 is a square, and 50 is divisible by a square (25).
Next in the sequence is 242, the first of four consecutive NSF (non-square-free) integers: 242 is divisible by 121 (11 squared); 243 is divisible by 9; 244 is divisible by 4; and 245 is divisible by 49. The sequence continues with the first of five consecutive NSF integers, the first of six, etc., up to the first of eighteen consecutive NSF integers, which seems to be the highest run found as of 2023. (There was a program available for finding more, but I have not tried it out.)
The OEIS has another version of this list, with a different number in the tenth position. This is because the smallest consecutive NSF integer run containing exactly ten integers is actually larger than the smallest containing eleven. The first OEIS entry therefore shows the smallest 11-integer run in both the tenth and eleventh spots, whereas the second entry shows the first run of exactly ten integers in the tenth spot.
That 11-integer run starts with the number 221,167,422. Here is the factorization of the numbers in that run:
| Number | Prime factorization | Contains |
|---|---|---|
| 221,167,422 | 2 × 35 × 7 × 65011 | 35 |
| 221,167,423 | 312 × 230143 | 312 |
| 221,167,424 | 26 × 3455741 | 26 |
| 221,167,425 | 3 × 52 × 23 × 128213 | 52 |
| 221,167,426 | 2 × 372 × 80777 | 372 |
| 221,167,427 | 132 × 29 × 45127 | 132 |
| 221,167,428 | 22 × 3 × 18430619 | 22 |
| 221,167,429 | 73 × 19 × 33937 | 73 |
| 221,167,430 | 2 × 5 × 112 × 47 × 3889 | 112 |
| 221,167,431 | 32 × 109 × 131 × 1721 | 32 |
| 221,167,432 | 23 × 2099 × 13171 | 23 |
Below is the prime factorization for the (first known) 18-integer run. I used this handy site to get the prime factorizations.
| Seq. | Number | Prime factorization | Contains |
|---|---|---|---|
| 1 | 125,781,000,834,058,568 | 23 × 283 × 503 × 647 × 887 × 192461 | 23 |
| 2 | 125,781,000,834,058,569 | 32 × 22176029 × 630215029 | 32 |
| 3 | 125,781,000,834,058,570 | 2 × 5 × 7 × 11 × 13 × 372 × 101 × 90877453 | 372 |
| 4 | 125,781,000,834,058,571 | 312 × 130885536768011 | 312 |
| 5 | 125,781,000,834,058,572 | 22 × 3 × 787 × 13318615081963 | 22 |
| 6 | 125,781,000,834,058,573 | 192 × 37277 × 9346884809 | 192 |
| 7 | 125,781,000,834,058,574 | 2 × 292 × 71 × 1053248152217 | 292 |
| 8 | 125,781,000,834,058,575 | 3 × 52 × 41 × 6983 × 5857710227 | 52 |
| 9 | 125,781,000,834,058,576 | 24 × 43 × 182821222142527 | 24 |
| 10 | 125,781,000,834,058,577 | 72 × 232 × 89 × 34267 × 1591099 | 72, 232 |
| 11 | 125,781,000,834,058,578 | 2 × 33 × 19483 × 119554370129 | 33 |
| 12 | 125,781,000,834,058,579 | 1392 × 6510066809899 | 1392 |
| 13 | 125,781,000,834,058,580 | 22 × 5 × 127 × 49520079068527 | 22 |
| 14 | 125,781,000,834,058,581 | 3 × 112 × 346504134529087 | 112 |
| 15 | 125,781,000,834,058,582 | 2 × 172 × 53 × 73 × 56245590151 | 172 |
| 16 | 125,781,000,834,058,583 | 132 × 1951 × 24181 × 15775997 | 132 |
| 17 | 125,781,000,834,058,584 | 23 × 3 × 7 × 449 × 1277 × 1305775531 | 23 |
| 18 | 125,781,000,834,058,585 | 5 × 1992 × 635241538517 | 1992 |
For this run, powers of two or greater appear for each of the first twelve prime numbers:
- 22+ appears in the 1st, 5th, 9th, 13th, and 17th numbers in the sequence.
- 32+ appears in the 2nd and 11th numbers.
- 52 appears in the 8th number.
- 72 appears in the 10th number.
- 112 appears in the 14th number.
- 132 appears in the 16th number.
- 172 appears in the 15th number.
- 192 appears in the 6th number.
- 232 appears in the 10th number.
- 292 appears in the 7th number.
- 312 appears in the 4th number.
- 372 appears in the 3rd number.
The other two squares that appear are 1392 and 1992.
WillNicholes.com 
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