## Thursday, November 24, 2016

### [ruzomkls] Smoother numbers

Given two numbers, which is smoother?  This is mostly an aesthetic question; there is no right answer.  Start by comparing their largest prime factor.  The one with the larger prime factor is less smooth.

If they are the same, we could compare their second largest prime factors, equivalently comparing the smoothness of the two numbers divided through once by their common largest prime factor.  Here is that ranking of smoothness of numbers 1 through 99:

1 2 4 8 16 32 64 3 6 12 24 48 96 9 18 36 72 27 54 81 5 10 20 40 80 15 30 60 45 90 25 50 75 7 14 28 56 21 42 84 63 35 70 49 98 11 22 44 88 33 66 99 55 77 13 26 52 39 78 65 91 17 34 68 51 85 19 38 76 57 95 23 46 92 69 29 58 87 31 62 93 37 74 41 82 43 86 47 94 53 59 61 67 71 73 79 83 89 97

Alternatively, if two numbers share the same largest prime factor, then declare the smaller number smoother.  Here is such a ranking for 1-99:

1 2 4 8 16 32 64 3 6 9 12 18 24 27 36 48 54 72 81 96 5 10 15 20 25 30 40 45 50 60 75 80 90 7 14 21 28 35 42 49 56 63 70 84 98 11 22 33 44 55 66 77 88 99 13 26 39 52 65 78 91 17 34 51 68 85 19 38 57 76 95 23 46 69 92 29 58 87 31 62 93 37 74 41 82 43 86 47 94 53 59 61 67 71 73 79 83 89 97

Inspiration is that smooth numbers might more memorable.  The perfect powers might be memorable 0 1 4 8 9 16 25 27 32 36 49 64 81.  Let the more important or more frequent public transit bus routes have the more memorable numbers.

Alternative or more complicated criteria possible.  Smoothness of exponents: 2^2^2^2 seems smoother than (say) 2^13.  64 seems smoother than 32.  Number of factors or divisors.  Squarefree numbers might be considered smoother, then avoiding multiple squared prime factors, then avoiding cubes, etc.