000RM.dvi

424 Digital sum and digital root

15.1 Digital sum sequences ..................

The digital sum of a positive integernis the sum of the digits ofn.We

denote this byd(n).

Given a positive integera, the digit sum sequenceS(a)=(an)is

defined recursively by

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

Here are the first few digit sum sequences:

S(1) 1, 2 , 4 , 8 , 16 , 23 , 28 , 38 , 49 , 62 , 70 , 77 , 91 , 101 , 103 , 107 ,...

S(3) 3, 6 , 12 , 15 , 21 , 24 , 30 , 33 , 39 , 51 , 57 , 69 , 84 , 96 , 111 , 114 ,...

S(5) 5, 10 , 11 , 13 , 17 , 25 , 32 , 37 , 47 , 58 , 71 , 79 , 95 , 109 , 119 , 130 ,...

S(7) 7, 14 , 19 , 29 , 40 , 44 , 52 , 59 , 73 , 83 , 94 , 107 , 115 , 122 , 127 , 137 ,...

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

Note that they are quite similar to the digital root sequences.

Show thatS(3) =R(3)andR(9) =S(9).

What is the smallest number that does not appear in any of these digit

sum sequences?

Find the first 10 terms of the digital sum sequence beginning with this

number.

20,22,26,34,41,46,56,67,80,88,104,109,119,130,134,142,...

Find the next smallest number which is not in any of the 6 digit sum

sequences and generate a new digit sum sequence from it.

31,35,43,50,55,65,76,89,106,113,118,128,139,152,160,167,...

There are infinitely many digit sum sequences because there are in-

finitely many numbers which are not of the formn+d(n).

The number 101 isn+d(n)forn=91and 100.

The number 101 traces back to 100, 86 which is a starter. It also

traces back to 91, and eventually 1.

Here are the numbers below 100 which are not of this form:

1 , 3 , 5 , 7 , 9 , 20 , 31 , 42 , 53 , 64 , 75 , 86 , 97.

An infinite sequence of “starters”: (^10) n 122 ,n≥ 1. Every number
n≤ (^10) n 114 hasn+d(n)≤ (^10) n 121 ; every numbern≥ (^10) n 115 has
d(n)≥ (^10) n 123.