23, 24 and 25 is one such set.
Divide the sum of the three consecutive integers by 3: 138/3 = 46. The smallest of these integers will be one less than 46 and the largest will be one more than 46, so the three consecutive integers will be 45, 46, and 47.
Divide the sum of the three consecutive odd integers by 3: -3 /3 = -1. The smallest of these integers will be two less than -1 and the largest will be two more than -1, so the three consecutive odd integers will be -3, -1, and +1.
Divide the sum of the three consecutive odd integers by 3: -273 /3 = -91. The smallest of these integers will be two less than -91 and the largest will be two more than -91, so the three consecutive odd integers will be -89, -91, and -93.
Divide the sum of the three consecutive odd integers by 3: -147 /3 = -49. The smallest of these integers will be two less than -49 and the largest will be two more than -49, so the three consecutive odd integers will be -47, -49, and -51.
Divide the sum of the three consecutive odd integers by 3: 45/3 = 15. The smallest of these integers will be two less than 15 and the largest will be two more than 15, so the three consecutive odd integers will be 13, 15, and 17.
Divide the sum of the three consecutive odd integers by 3: 402/3 = 134. The smallest of these integers will be two less than 134 and the largest will be two more than 134, so the three consecutive odd integers will be 132, 134, and 136.
Divide the sum of the three consecutive odd integers by 3: 93 / 3 = 31. The smallest of these integers will be two less than 31 and the largest will be two more than 31, so the three consecutive odd integers will be 29, 31, and 33.
Divide the sum of the three consecutive odd integers by 3: 225/3 = 75. The smallest of these integers will be two less than 75 and the largest will be two more than 75, so the three consecutive odd integers will be 73, 75, and 77.
Divide the sum of the three consecutive odd integers by 3: 51 / 3 = 17. The smallest of these integers will be two less than 17 and the largest will be two more than 17, so the three consecutive odd integers will be 15, 17, and 19.
Divide the sum of the three consecutive odd integers by 3: 159/3 = 53. The smallest of these integers will be two less than 53 and the largest will be two more than 53, so the three consecutive odd integers will be 51, 53, and 55.
The only proper factor of 51 that is less than 1/3 of 51 is 3. Divide the sum of three consecutive odd integers by 3: 51/3 = 17. The smallest of these integers will be two less than 17 and the largest will be two more than 17, so the three consecutive odd integers having a sum of 51 are 15, 17, and 19.
There are only 3 sets of consecutive even integers less than 12, you can work it out...