# Round Numbers

When someone says “a nice round number, like 1000,” what are they really saying? What about a number makes it round?

The most reasonable explanation is probably that it is has very few non-zero digits, thus giving it properties like being easy to remember, easy to put in bundles, and easy to divide up. But if one thinks a bit deeper about these qualities of round numbers, one can find that typically-considered round numbers are really not the optimally round ones to find.

1000, for example, is only round because we use base ten, which is only because we happen to have ten fingers. If would also be round if we used base five or twenty, not so round if we used base two, and not round at all if we used base nine (not that nine is that reasonable of a base to use).

We can satisfy properties of being easy to put in bundles and being easy to divide up much better, and be more correct about few non-zero digits by a theoretical viewpoint, by looking for numbers that are the original definition of round…in a particularly large number of bases: numbers with many factors. How would we elegantly define this type of having a lot of factors?

Here, I will propose a new definition for a round number: it is a number $n$ with the property that for all pairs of primes $p$ and $q$, with $e_p$ and $e_q$ being the respective powers of $p$ and $q$ in the prime factorization of $n$, if $q>p$, then $e_q \leq e_p$, or in other words, $e_2 \geq e_3 \geq e_5 \geq e_7 \geq e_{11} \geq ...$.

Here are some properties of round numbers as defined, with all variables in the statements below being positive integers:

-1 is the only odd round number.
-If $n$ is round, then $n^m$ is round.
-If $\sqrt[m]{n}$ is an integer, then if $n$ is round, then $\sqrt[m]{n}$ is round.
$10^n$ is not round.
-All round numbers are abundant, except for 6 and the powers of 2.

We can also define barely round numbers as numbers that just barely qualify for being round: the boundary cases where all prime factors are 2 or exactly one factor of each prime factor involved exists, namely, the powers of two and the primordials. Note that if a round number is not barely round, then it is necessarily abundant.

Here is a list of round numbers less than 65537, by the new definition:

1
2
4
6
8
12
16
24
30
32
36
48
60
64
72
96
120
128
144
180
192
210
216
240
256
288
360
384
420
432
480
512
576
720
768
840
864
900
960
1024
1080
1152
1260
1296
1440
1536
1680
1728
1800
1920
2048
2160
2304
2310
2520
2592
2880
3072
3360
3456
3600
3840
4096
4320
4608
4620
5040
5184
5400
5760
6144
6300
6480
6720
6912
7200
7560
7680
7776
8192
8640
9216
9240
10080
10368
10800
11520
12288
12600
12960
13440
13824
13860
14400
15120
15360
15552
16384
17280
18432
18480
20160
20736
21600
23040
24576
25200
25920
26880
27000
27648
27720
28800
30030
30240
30720
31104
32400
32768
34560
36864
36960
37800
38880
40320
41472
43200
44100
45360
46080
46656
49152
50400
51840
53760
54000
55296
55440
57600
60060
60480
61440
62208
64800
65536

This post has 430 words, or, when rounded, 432.

## 5 thoughts on “Round Numbers”

1. This is pretty cool! I’d just like to add that when you apply Euler’s Totient Function on any of the given round numbers you get another round number…I haven’t been able to prove that’s always true, but it seems somewhat feasible.

1. No, this is false. phi(2*3*5*7*11*13*17*19*23) is not “round”, as it is divisible by 11 but not 7.

1. Ah, nicely constructed! Using one prime number being one more than a multiple of another. These cases do seem very rare, though. Would you like to give them a name?

2. I googled “round numbers” trying to get the joke in xkcd #1000, but didn’t find you until I searched for “list of round numbers.” I find your definition more satisfying than the rather vague one cited on Wikipedia (which doesn’t appear to be very authoritative).

Just curious: how did you decide how many to include in your list? Why not stop at the 144th?

1. I decided to stop at a factorion.