Digit sum

In

${\displaystyle 9045}$

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

Let ${\displaystyle n}$ be a natural number. We define the digit sum for base ${\displaystyle b>1}$, ${\displaystyle F_{b}:\mathbb {N} \rightarrow \mathbb {N} }$ to be the following:

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

where ${\displaystyle k=\lfloor \log _{b}{n}\rfloor }$ is one less than the number of digits in the number in base ${\displaystyle b}$, and

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

is the value of each digit of the number.

For example, in

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

For any two bases ${\displaystyle 2\leq b_{1}<b_{2}}$ and for sufficiently large natural numbers ${\displaystyle n,}$

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

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.^[2]

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

The amount of n-digit numbers with digit sum q can be calculated using:

${\displaystyle f(n,q)={\begin{cases}1&{\text{if }}n=1\\f(n,9n-q+1)&{\text{if }}q>\lceil {\frac {9n}{2}}\rceil \\\sum _{i=\max(q-9,1)}^{q}f(n-1,i)&{\text{otherwise}}\end{cases}}}$

The concept of a decimal digit sum is closely related to, but not the same as, the

Harshad numbers are defined in terms of divisibility by their digit sums, and Smith numbers are defined by the equality of their digit sums with the digit sums of their prime factorizations.

• Arithmetic dynamics
• Casting out nines
• Checksum
• Digital root
• Hamming weight
• Harshad number
• Perfect digital invariant
• Sideways sum
• Smith number
• Sum-product number