I first noticed these patterns while playing around with a calculator as a child. I still do this occasionally, so this list has expanded over time.
111/37 = 3
222/37 = 6
333/37 = 9
444/37 = 12
555/37 = 15
666/37 = 18
777/37 = 21
888/37 = 24
999/37 = 27
Also notice that the sum of the digits adds up to the result (e.g. 1+1+1 = 3).
The sum of the digits in the product yields the original number:
11*37 = 407
4+0+7 = 11
13*37 = 481
4+8+1 = 13
17*37 = 629
6+2+9 = 17
For single digit primes (except for 3, see the triplets above) this property can be seen by summing the digits twice:
2*37 = 74
7+4 = 11
1+1 = 2
5*37 = 185
1+8+5 = 14
1+4 = 5
7*37 = 259
2+5+9 = 16
1+6 = 7
The sum of the digits in the product multiplied by 37 again produces a permutation of the original product:
23*37 = 851
8+5+1 = 14
14*37 = 518
The sum of the digits in the product is the reverse of the original number. Notice that this works with all two digit primes ending in 1:
11*37 = 407
4+0+7 = 11
31*37 = 1147
1+1+4+7 = 13
41*37 = 1517
1+5+1+7 = 14
61*37 = 2257
2+2+5+7 = 16
71*37 = 2627
2+6+2+7 = 17
91*37 = 3367
3+3+6+7 = 19
The sum of the digits of the product equals the sum of the digits of the original number:
19*37 = 703
7+0+3 = 1+9
29*37 = 1073
1+0+7+3 = 2+9
79*37 = 2923
2+9+2+3 = 7+9
83*37 = 3071
3+0+7+1 = 8+3
89*37 = 3293
3+2+9+3 = 8+9
This property also holds for some of the multiples of these primes, for example:
19*5 = 95
95*37 = 3515
3+5+1+5 = 9+5
29*2 = 58
58*37 = 2146
2+1+4+6 = 5+8
19*37 = 703
703*37 = 26011
2+6+0+1+1 = 7+0+3