000RM.dvi

15.4 Digital root sequences 427

15.4 Digital root sequences..................

A digit root sequence(an)is defined recursively by

an+1=an+D(an),a 1 =a.

The digital root sequence beginning with 1 is

1 , 2 , 4 , 8 , 16 , 23 , 28 , 29 , 31 , 35 , 43 , 50 , 55 , 56 , 58 , 62 , 70 ,...

Obviously, the digital root sequence beginning with any of these terms is
a subsequence of this. By taking a term not in this sequence we generate
a new one. Here are five digital root sequences with initial terms 1, 3, 5,
7, 9.

R(1) 1, 2 , 4 , 8 , 16 , 23 , 28 , 29 , 31 , 35 , 43 , 50 , 55 , 56 , 58 , 62 , 70 , 77 , 82 , 83 , 85 , 89 , 97 , 104 ,...

R(3) 3, 6 , 12 , 15 , 21 , 24 , 30 , 33 , 39 , 42 , 48 , 51 , 57 , 60 , 66 , 69 , 75 , 78 , 84 , 87 , 93 , 96 , 102 ,...

R(5) 5, 10 , 11 , 13 , 17 , 25 , 32 , 37 , 38 , 40 , 44 , 52 , 59 , 64 , 65 , 67 , 71 , 79 , 86 , 91 , 92 , 94 , 98 , 106 ,...

R(7) 7, 14 , 19 , 20 , 22 , 26 , 34 , 41 , 46 , 47 , 49 , 53 , 61 , 68 , 73 , 74 , 76 , 80 , 88 , 95 , 100 ,...

R(9) 9, 18 , 27 , 36 , 45 , 54 , 63 , 72 , 81 , 90 , 99 , 108 , 117 , 126 , 135 , 144 , 153 ,...

Theorem 15.2(Kumar).1.The digital roots of these sequences are
periodic.

2.These five sequences partition the natural numbers.

The digital roots of these sequences are

R 1 1 , 2 , 4 , 8 , 7 , 5

R 3 3 ,^6

R 5 5 , 1 , 2 , 4 , 8 , 7

R 7 7 , 5 , 1 , 2 , 4 , 8

R 9 9