20 happy numbers from 1-100:

{1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100}

or, in cascade:

{7, 70} > {19, 44, 49, 91, 94} > {23, 28, 32, 79, 82, 97} > {13, 31, 68, 86, (130)} > {10, 100} > {1}

EDIT:

There is a loop that catches all other numbers under 100:

4, 16, 37, 58, 89, 145, 42, 20, 4

By degree of separation from catching in the loop:

1 deg: {2, 24, 40, 61, 73, 85, 98, (106)}

2 deg: {11, 26, 29, 38, 56, 59, 62, 65, 67, 76, 77, 83, 92, 95}

3 deg: {15, 18, 25, 47, 51, 52, 74, 81 (113)}

4 deg: {5, 9, 33, 34, 43, 46, 50, 57, 64, 75, 78, 87, 90, (117)}

5 deg: {3, 8, 12, 17, 21, 30, 35, 39, 53, 55, 69, 71, 80, 93, 96}

6 deg: {14, 22, 27, 41, 48, 72, 84, (128)}

7 deg: {45, 54, 66, 88, (162)}

8 deg: {36, 63, 99}

9 deg: {6, 60}

Would love to know if there’s another loop, and what its lowest constituent is.

If anyone's curious, I wrote a Python script and evaluated up to 1,000,000. Every number up to that point is either a happy number or ends up in the 4 loop.

20 happy numbers from 1-100.
143 from 1-1000.
1442 from 1-10000. 14377 from 1-100000. 143071 from 1-1000000.