Digitaddition

A digitaddition of a nonnegative integer is the number obtained by adding the sum of its decimal digits to . The number is called a digitaddition generator of . A number can have more than one digitaddition generator. If a number has no digitaddition generator, it is called a self number. In a finite sequence of repeated digitadditions, the sum of the digit sums of all terms equals the last term minus the first plus the digit sum of the last.

Repeated digitaddition strictly increases every positive integer. Repeatedly taking only the digit sum instead eventually gives the digital root. The base-10 digital roots of the first few positive integers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, ... (OEIS A010888).

The related -digitaddition iteration replaces a positive integer by the sum of the th powers of its decimal digits, for a positive integer . Every starting number eventually reaches a periodic sequence. For example, the 2-digitaddition sequence for begins 2, , , , , , , and so on.

If equals the sum of the th powers of its digits, then is a fixed point of this iteration. It is a narcissistic number when also equals the number of its decimal digits. The smallest member of a cycle of length greater than 1 is called a recurring digital invariant.

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, ... (OEIS A031176). Similarly, the numbers that correspond to the beginning of the eventually periodic part of a 2-digitaddition sequence are given by 1, 4, 37, 4, 89, 89, 1, 89, 37, 1, 4, ... (OEIS A103369).

The possible periods for -digitadditions are summarized in the following table, together with digitadditions for the first few integers and the corresponding sequence numbers. Some periods do not show up for a long time. For example, a period-6 10-digitaddition does not occur until the number 266.

	OEIS	s	-digitadditions
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, 3, 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, 2, 6, 7, 17, 81, 123	1, 17, 123, 17, 17, 123, 123, 123, 123, 1, 17, 123, 17, ...

The numbers having period-1 2-digitaddition sequences are also called happy numbers, the first few of which are 1, 7, 10, 13, 19, 23, 28, 31, 32, ... (OEIS A007770).

The first few numbers having period -digitadditions are summarized in the following table.

		OEIS	members
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		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, 11458, 11485, ...
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	3		111348, ...
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		1, 10, 100, 1000, 10000, 100000, ...
10	2		34668, ...
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, ...

Every power of 10 (OEIS A011557) reaches the fixed point 1 for every positive .