## February 18th, 2019

### When the square root of n is less than or equal to n's number of divisors

Just in case you ever needed to know, there are 55 positive integers whose square root √n is less than or equal to their number of divisors τ(n).

√n < τ(n) in 53 cases, and in two cases √n = τ(n): 1 and 9.

They are:
 n τ(n) √n 1 1 1 2 2 1.41 3 2 1.73 4 3 2 6 4 2.45 8 4 2.83 9 3 3 10 4 3.16 12 6 3.46 14 4 3.74 15 4 3.87 16 5 4 18 6 4.24 20 6 4.47 24 8 4.90 28 6 5.30 30 8 5.48 32 6 5.66 36 9 6 40 8 6.32 42 8 6.48 48 10 6.93 54 8 7.35 56 8 7.48 60 12 7.75 72 12 8.49 80 10 8.94 84 12 9.17 90 12 9.49 96 12 9.80 108 12 10.39 120 16 10.95 126 12 11.22 132 12 11.49 140 12 11.83 144 15 12 168 16 12.96 180 18 13.42 192 14 13.86 210 16 14.49 216 16 14.70 240 20 15.49 252 18 15.87 288 18 16.97 300 18 17.32 336 20 18.33 360 24 18.97 420 24 20.49 480 24 21.91 504 24 22.45 540 24 23.24 720 30 26.83 840 32 28.98 1260 36 35.50

Or to put it another way:

 n τ(n) √n 1 1 1 2 2 1.41 3 1.73 4 3 2 9 3 6 4 2.45 8 2.83 10 3.16 14 3.74 15 3.87 16 5 4 12 6 3.46 18 4.24 20 4.47 28 5.29 32 5.66 24 8 4.90 30 5.48 40 6.32 42 6.48 54 7.35 56 7.48 36 9 6 48 10 6.93 80 8.94 60 12 7.75 72 8.49 84 9.17 90 9.49 96 9.80 108 10.39 126 11.22 132 11.49 140 11.83 192 14 13.86 144 15 12 120 16 10.95 168 12.96 210 14.49 216 14.70 180 18 13.42 252 15.87 288 16.97 300 17.32 240 20 15.49 336 18.33 360 24 18.97 420 20.49 480 21.91 504 22.45 540 23.24 720 26.83 840 32 28.98 1260 36 35.50

The number whose square root is smallest relative to its number of divisors is 12, whose 6 divisors are almost twice its square root of 3.45.

There, aren't you glad I told you?

Edited to add: Looks like I miscounted and there are only 54.