從一個 整數 
開始，稱為數字加法生成元。將數字加法生成元的各位數字之和加到該數上，得到數字加法 
。一個數字可以有多個數字加法生成元。如果一個數字沒有數字加法生成元，則稱為自守數。數字加法序列中所有數字之和等於最後一項減去第一項，再加上最後一項的數字之和。
如果在 
上執行數字加法過程，得到其數字加法 
，在 
上執行得到 
，等等，最終會得到一個單數字，稱為 
的數字根。前幾個整數的數字根是 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, ... (OEIS A010888)。
如果將該過程推廣，使得重複加一個數字的各位數字的第 
次方（而不是一次方），那麼對於任何給定的起始數字 
，最終都會得到一個週期序列。例如，
的 2-數字加法序列為 2, 
, 
, 
, 
, 
, 
, 等等。
如果原始數字 
等於其各位數字的第 
次方之和（即，數字加法序列的長度為 2），則 
稱為自戀數。如果原始數字是重複 
-數字加法最終得到的週期序列中最小的數字，則稱為迴圈數字不變式。自戀數和迴圈數字不變式都相對稀少。
重複 2-數字加法的唯一可能週期是 1 和 8，前幾個正整數的週期是 1, 8, 8, 8, 8, 8, 1, 8, 8, 1, ... (OEIS A031176)。類似地，與 2-數字加法序列最終週期部分的開頭對應的數字由 1, 4, 37, 4, 89, 89, 1, 89, 37, 1, 4, ... (OEIS A103369) 給出。
下表總結了 
-數字加法的可能週期 
，以及前幾個整數的數字加法和相應的序列號。有些週期很長時間才出現。例如，週期為 6 的 10-數字加法直到數字 266 才出現。
 | OEIS | 週期  s |  -數字加法 |
| 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, ... |
週期為 1 的 2-數字加法序列的數字也稱為快樂數，前幾個快樂數是 1, 7, 10, 13, 19, 23, 28, 31, 32, ... (OEIS A007770)。
下表總結了週期為 
的 
-數字加法的前幾個數字。
 |  | OEIS | 成員 |
| 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, ... |
