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 with the property that for all pairs of primes and , with and being the respective powers of and in the prime factorization of , if , then , or in other words, .

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 is round, then is round.

-If is an integer, then if is round, then is round.

– 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.

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.

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.

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?

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?

I decided to stop at a factorion.