从一个 整数 
开始，称为数字加法生成元。将数字加法生成元的各位数字之和加到该数上，得到数字加法 
。一个数字可以有多个数字加法生成元。如果一个数字没有数字加法生成元，则称为自守数。数字加法序列中所有数字之和等于最后一项减去第一项，再加上最后一项的数字之和。
如果在 
上执行数字加法过程，得到其数字加法 
，在 
上执行得到 
，等等，最终会得到一个单数字，称为 
的数字根。前几个整数的数字根是 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, ... (OEIS A010888)。
如果将该过程推广，使得重复加一个数字的各位数字的第 
次方（而不是一次方），那么对于任何给定的起始数字 
，最终都会得到一个周期序列。例如，
的 2-数字加法序列为 2, 
, 
, 
, 
, 
, 
, 等等。
如果原始数字 
等于其各位数字的第 
次方之和（即，数字加法序列的长度为 2），则 
称为自恋数。如果原始数字是重复 
-数字加法最终得到的周期序列中最小的数字，则称为循环数字不变式。自恋数和循环数字不变式都相对稀少。
重复 2-数字加法的唯一可能周期是 1 和 8，前几个正整数的周期是 1, 8, 8, 8, 8, 8, 1, 8, 8, 1, ... (OEIS A031176)。类似地，与 2-数字加法序列最终周期部分的开头对应的数字由 1, 4, 37, 4, 89, 89, 1, 89, 37, 1, 4, ... (OEIS A103369) 给出。
下表总结了 
-数字加法的可能周期 
，以及前几个整数的数字加法和相应的序列号。有些周期很长时间才出现。例如，周期为 6 的 10-数字加法直到数字 266 才出现。
 | OEIS | 周期  s |  -数字加法 |
| 2 | A031176 | 1, 8 | 1, 8, 8, 8, 8, 8, 1, 8, 8, 1, ... |
| 3 | A031178 | 1, 2, 3 | 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, ... |
| 4 | A031182 | 1, 2, 7 | 1, 7, 7, 7, 7, 7, 7, 7, 7, 1, 7, 1, 7, 7, ... |
| 5 | A031186 | 1, 2, 4, 6, 10, 12, 22, 28 | 1, 12, 22, 4, 10, 22, 28, 10, 22, 1, ... |
| 6 | A031195 | 1, 2, 3, 4, 10, 30 | 1, 10, 30, 30, 30, 10, 10, 10, 3, 1, 10, ... |
| 7 | A031200 | 1, 2, 3, 6, 12, 14, 21, 27, 30, 56, 92 | 1, 92, 14, 30, 92, 56, 6, 92, 56, 1, 92, 27, ... |
| 8 | A031211 | 1, 25, 154 | 1, 25, 154, 154, 154, 154, 25, 154, 154, 1, 25, 154, 154, 1, ... |
| 9 | A031212 | 1, 2, 3, 4, 8, 10, 19, 24, 28, 30, 80, 93 | 1, 30, 93, 1, 19, 80, 4, 30, 80, 1, 30, 93, 4, 10, ... |
| 10 | A031213 | 1, 6, 7, 17, 81, 123 | 1, 17, 123, 17, 17, 123, 123, 123, 123, 1, 17, 123, 17, ... |
周期为 1 的 2-数字加法序列的数字也称为快乐数，前几个快乐数是 1, 7, 10, 13, 19, 23, 28, 31, 32, ... (OEIS A007770)。
下表总结了周期为 
的 
-数字加法的前几个数字。
 |  | OEIS | 成员 |
| 2 | 1 | A007770 | 1, 7, 10, 13, 19, 23, 28, 31, 32, ... |
| 2 | 8 | A031177 | 2, 3, 4, 5, 6, 8, 9, 11, 12, 14, 15, ... |
| 3 | 1 | A031179 | 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, ... |
| 3 | 2 | A031180 | 49, 94, 136, 163, 199, 244, 316, ... |
| 3 | 3 | A031181 | 4, 13, 16, 22, 25, 28, 31, 40, 46, ... |
| 4 | 1 | A031183 | 1, 10, 12, 17, 21, 46, 64, 71, 100, ... |
| 4 | 2 | A031184 | 66, 127, 172, 217, 228, 271, 282, ... |
| 4 | 7 | A031185 | 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, ... |
| 5 | 1 | A031187 | 1, 10, 100, 145, 154, 247, 274, ... |
| 5 | 2 | A031188 | 133, 139, 193, 199, 226, 262, ... |
| 5 | 4 | A031189 | 4, 37, 40, 55, 73, 124, 142, ... |
| 5 | 6 | A031190 | 16, 61, 106, 160, 601, 610, 778, ... |
| 5 | 10 | A031191 | 5, 8, 17, 26, 35, 44, 47, 50, 53, ... |
| 5 | 12 | A031192 | 2, 11, 14, 20, 23, 29, 32, 38, 41, ... |
| 5 | 22 | A031193 | 3, 6, 9, 12, 15, 18, 21, 24, 27, ... |
| 5 | 28 | A031194 | 7, 13, 19, 22, 25, 28, 31, 34, 43, ... |
| 6 | 1 | A011557 | 1, 10, 100, 1000, 10000, 100000, ... |
| 6 | 2 | A031357 | 3468, 3486, 3648, 3684, 3846, ... |
| 6 | 3 | A031196 | 9, 13, 31, 37, 39, 49, 57, 73, 75, ... |
| 6 | 4 | A031197 | 255, 466, 525, 552, 646, 664, ... |
| 6 | 10 | A031198 | 2, 6, 7, 8, 11, 12, 14, 15, 17, 19, ... |
| 6 | 30 | A031199 | 3, 4, 5, 16, 18, 22, 29, 30, 33, ... |
| 7 | 1 | A031201 | 1, 10, 100, 1000, 1259, 1295, ... |
| 7 | 2 | A031202 | 22, 202, 220, 256, 265, 526, 562, ... |
| 7 | 3 | A031203 | 124, 142, 148, 184, 214, 241, 259, ... |
| 7 | 6 | 7, 70, 700, 7000, 70000, 700000, ... | |
| 7 | 12 | A031204 | 17, 26, 47, 59, 62, 71, 74, 77, 89, ... |
| 7 | 14 | A031205 | 3, 30, 111, 156, 165, 249, 294, ... |
| 7 | 21 | A031206 | 19, 34, 43, 91, 109, 127, 172, 190, ... |
| 7 | 27 | A031207 | 12, 18, 21, 24, 39, 42, 45, 54, 78, ... |
| 7 | 30 | A031208 | 4, 13, 16, 25, 28, 31, 37, 40, 46, ... |
| 7 | 56 | A031209 | 6, 9, 15, 27, 33, 36, 48, 51, 57, ... |
| 7 | 92 | A031210 | 2, 5, 8, 11, 14, 20, 23, 29, 32, 35, ... |
| 8 | 1 | 1, 10, 14, 17, 29, 37, 41, 71, 73, ... | |
| 8 | 25 | 2, 7, 11, 15, 16, 20, 23, 27, 32, ... | |
| 8 | 154 | 3, 4, 5, 6, 8, 9, 12, 13, 18, 19, ... | |
| 9 | 1 | 1, 4, 10, 40, 100, 400, 1000, 1111, ... | |
| 9 | 2 | 127, 172, 217, 235, 253, 271, 325, ... | |
| 9 | 3 | 444, 4044, 4404, 4440, 4558, ... | |
| 9 | 4 | 7, 13, 31, 67, 70, 76, 103, 130, ... | |
| 9 | 8 | 22, 28, 34, 37, 43, 55, 58, 73, 79, ... | |
| 9 | 10 | 14, 38, 41, 44, 83, 104, 128, 140, ... | |
| 9 | 19 | 5, 26, 50, 62, 89, 98, 155, 206, ... | |
| 9 | 24 | 16, 61, 106, 160, 337, 373, 445, ... | |
| 9 | 28 | 19, 25, 46, 49, 52, 64, 91, 94, ... | |
| 9 | 30 | 2, 8, 11, 17, 20, 23, 29, 32, 35, ... | |
| 9 | 80 | 6, 9, 15, 18, 24, 33, 42, 48, 51, ... | |
| 9 | 93 | 3, 12, 21, 27, 30, 36, 39, 45, 54, ... | |
| 10 | 1 | A011557 | 1, 10, 100, 1000, 10000, 100000, ... |
| 10 | 6 | 266, 626, 662, 1159, 1195, 1519, ... | |
| 10 | 7 | 46, 58, 64, 85, 122, 123, 132, ... | |
| 10 | 17 | 2, 4, 5, 11, 13, 20, 31, 38, 40, ... | |
| 10 | 81 | 17, 18, 37, 71, 73, 81, 107, 108, ... | |
| 10 | 123 | 3, 6, 7, 8, 9, 12, 14, 15, 16, 19, ... |
