Start with an Integer , known as the Generator. Add the Sum of the Generator's digits to the Generator to obtain the digitaddition . A number can have more than one Generator. If a number has no Generator, it is called a Self Number. The sum of all numbers in a digitaddition 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., a
single-digit number, known as the Digital Root of , is eventually obtained. The digital roots of the first few
integers 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 is equal to the sum of the th powers of its digits, it is called a Narcissistic Number. If the original number is the smallest number in the eventually periodic sequence of numbers in the repeated -digitadditions, it is called a Recurring Digital Invariant. Both Narcissistic Numbers and Recurring Digital 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 few numbers 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, ... |

© 1996-9

1999-05-24