The sequence https://oeis.org/A007953 is the sum of the digits of n.
1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,10,2,3...

Most terms of the sequence are 1 more than the previous term but some are not. The first two are shown in bold above.

Create the sequence of terms of A007953 that are not 1 more than the previous term.

Prove or disprove: This new sequence is the same as the original sequence.

Consecutive numbers in this sequence can only decrease when the ones digit goes from 9 to 0, and they will always decrease when this happens.

The sum of digits after the decrease, then, is for a number whose last digit is zero, and the series is for all such numbers.

Since the final digit of the numbers in this series is 0, it contributes, well, 0 to the sum of the digits, so all of the numbers in the sequence can be divided by 10 without changing the sums. But dividing every 10th number by 10 gives back the sequence 1..n itself, and so the sum of digits of this series is the same as the original.

Posted by Paul on 2014-05-14 22:44:11