Digit sum

In mathematics, the digit sum of a natural number in a given number base is the sum of all its digits. For example, the digit sum of the decimal number

9045

{\displaystyle 9045}

would be

9

+

0

+

4

+

5

=

18.

{\displaystyle 9+0+4+5=18.}

Definition

Let

n

{\displaystyle n}

be a natural number. We define the digit sum for base

b

>

1

{\displaystyle b>1}

,

F

b

:

N

→

N

{\displaystyle F_{b}:\mathbb {N} \rightarrow \mathbb {N} }

to be the following:

F

b

(

n

)

=

∑

i

=

0

k

d

i

{\displaystyle F_{b}(n)=\sum _{i=0}^{k}d_{i}}

where

k

=

⌊

log

b

⁡

n

⌋

{\displaystyle k=\lfloor \log _{b}{n}\rfloor }

is one less than the number of digits in the number in base

b

{\displaystyle b}

, and

d

i

=

n

mod

b

i

+

1

−

n

mod

b

i

b

i

{\displaystyle d_{i}={\frac {n{\bmod {b^{i+1}}}-n{\bmod {b}}^{i}}{b^{i}}}}

is the value of each digit of the number.

For example, in base 10, the digit sum of 84001 is

F

10

(

84001

)

=

8

+

4

+

0

+

0

+

1

=

13.

{\displaystyle F_{10}(84001)=8+4+0+0+1=13.}

For any two bases

2

≤

b

1

<

b

2

{\displaystyle 2\leq b_{1}<b_{2}}

and for sufficiently large natural numbers

n

,

{\displaystyle n,}

∑

k

=

0

n

F

b

1

(

k

)

<

∑

k

=

0

n

F

b

2

(

k

)

.

{\displaystyle \sum _{k=0}^{n}F_{b_{1}}(k)<\sum _{k=0}^{n}F_{b_{2}}(k).}

The sum of the base 10 digits of the integers 0, 1, 2, ... is given by OEIS: A007953 in the On-Line Encyclopedia of Integer Sequences. Borwein & Borwein (1992) use the generating function of this integer sequence (and of the analogous sequence for binary digit sums) to derive several rapidly converging series with rational and transcendental sums.

Extension to negative integers

The digit sum can be extended to the negative integers by use of a signed-digit representation to represent each integer.