Digitaddition
Start with an Integer 
, known as the Generator. Add the Sum of theGenerator's digits to the Generator to obtain thedigitaddition 
. A number can have more than one Generator. If a number has noGenerator, it is called a Self Number. The sum of all numbers in adigitaddition series is given by the last term minus the first plus the sum of the Digits of the last.
If the digitaddition process is performed on to yield its digitaddition 
, on to yield 
, etc., asingle-digit number, known as the Digital Root of , is eventually obtained. The digital roots of the first fewintegers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 9, 1, ... (Sloane's A010888).
If the process is generalized so that the 
th (instead of first) powers of the digits of a number are repeatedly added,a periodic sequence of numbers is eventually obtained for any given starting number . If the original number isequal to the sum of the th powers of its digits, it is called a Narcissistic Number. If the original number isthe smallest number in the eventually periodic sequence of numbers in the repeated -digitadditions, it is called aRecurring Digital Invariant. Both Narcissistic Numbers and RecurringDigital Invariants are relatively rare.
The only possible periods for repeated 2-digitadditions are 1 and 8, and the periods of the first few positive integers are 1,8, 8, 8, 8, 8, 1, 8, 8, 1, .... The possible periods 
for -digitadditions are summarized in the following table,together with digitadditions for the first few integers and the corresponding sequence numbers.
| Sloane | s | -Digitadditions |
| 2 | Sloane's A031176 | 1, 8 | 1, 8, 8, 8, 8, 8, 1, 8, 8, 1, ... |
| 3 | Sloane's A031178 | 1, 2, 3 | 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, ... |
| 4 | Sloane's A031182 | 1, 2, 7 | 1, 7, 7, 7, 7, 7, 7, 7, 7, 1, 7, 1, 7, 7, ... |
| 5 | Sloane's A031186 | 1, 2, 4, 6, 10, 12, 22, 28 | 1, 12, 22, 4, 10, 22, 28, 10, 22, 1, ... |
| 6 | Sloane's A031195 | 1, 2, 3, 4, 10, 30 | 1, 10, 30, 30, 30, 10, 10, 10, 3, 1, 10, ... |
| 7 | Sloane's 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 | Sloane's A031211 | 1, 25, 154 | 1, 25, 154, 154, 154, 154, 25, 154, 154, 1, 25, 154, 154, 1, ... |
| 9 | Sloane's 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 | Sloane's A031213 | 1, 6, 7, 17, 81, 123 | 1, 17, 123, 17, 17, 123, 123, 123, 123, 1, 17, 123, 17 ... |
The numbers having period-1 2-digitaded sequences are also called Happy Numbers. The first fewnumbers having period -digitadditions are summarized in the following table, together with their sequence numbers.
| | Sloane | Members |
| 2 | 1 | Sloane's A007770 | 1, 7, 10, 13, 19, 23, 28, 31, 32, ... |
| 2 | 8 | Sloane's A031177 | 2, 3, 4, 5, 6, 8, 9, 11, 12, 14, 15, ... |
| 3 | 1 | Sloane's A031179 | 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, ... |
| 3 | 2 | Sloane's A031180 | 49, 94, 136, 163, 199, 244, 316, ... |
| 3 | 3 | Sloane's A031181 | 4, 13, 16, 22, 25, 28, 31, 40, 46, ... |
| 4 | 1 | Sloane's A031183 | 1, 10, 12, 17, 21, 46, 64, 71, 100, ... |
| 4 | 2 | Sloane's A031184 | 66, 127, 172, 217, 228, 271, 282, ... |
| 4 | 7 | Sloane's A031185 | 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, ... |
| 5 | 1 | Sloane's A031187 | 1, 10, 100, 145, 154, 247, 274, ... |
| 5 | 2 | Sloane's A031188 | 133, 139, 193, 199, 226, 262, ... |
| 5 | 4 | Sloane's A031189 | 4, 37, 40, 55, 73, 124, 142, ... |
| 5 | 6 | Sloane's A031190 | 16, 61, 106, 160, 601, 610, 778, ... |
| 5 | 10 | Sloane's A031191 | 5, 8, 17, 26, 35, 44, 47, 50, 53, ... |
| 5 | 12 | Sloane's A031192 | 2, 11, 14, 20, 23, 29, 32, 38, 41, ... |
| 5 | 22 | Sloane's A031193 | 3, 6, 9, 12, 15, 18, 21, 24, 27, ... |
| 5 | 28 | Sloane's A031194 | 7, 13, 19, 22, 25, 28, 31, 34, 43, ... |
| 6 | 1 | Sloane's A011557 | 1, 10, 100, 1000, 10000, 100000, ... |
| 6 | 2 | Sloane's A031357 | 3468, 3486, 3648, 3684, 3846, ... |
| 6 | 3 | Sloane's A031196 | 9, 13, 31, 37, 39, 49, 57, 73, 75, ... |
| 6 | 4 | Sloane's A031197 | 255, 466, 525, 552, 646, 664, ... |
| 6 | 10 | Sloane's A031198 | 2, 6, 7, 8, 11, 12, 14, 15, 17, 19, ... |
| 6 | 30 | Sloane's A031199 | 3, 4, 5, 16, 18, 22, 29, 30, 33, ... |
| 7 | 1 | Sloane's A031201 | 1, 10, 100, 1000, 1259, 1295, ... |
| 7 | 2 | Sloane's A031202 | 22, 202, 220, 256, 265, 526, 562, ... |
| 7 | 3 | Sloane's A031203 | 124, 142, 148, 184, 214, 241, 259, ... |
| 7 | 6 | 7, 70, 700, 7000, 70000, 700000, ... | |
| 7 | 12 | Sloane's A031204 | 17, 26, 47, 59, 62, 71, 74, 77, 89, ... |
| 7 | 14 | Sloane's A031205 | 3, 30, 111, 156, 165, 249, 294, ... |
| 7 | 21 | Sloane's A031206 | 19, 34, 43, 91, 109, 127, 172, 190, ... |
| 7 | 27 | Sloane's A031207 | 12, 18, 21, 24, 39, 42, 45, 54, 78, ... |
| 7 | 30 | Sloane's A031208 | 4, 13, 16, 25, 28, 31, 37, 40, 46, ... |
| 7 | 56 | Sloane's A031209 | 6, 9, 15, 27, 33, 36, 48, 51, 57, ... |
| 7 | 92 | Sloane's 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 | Sloane's 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, ... |
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