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 formula_1 would be formula_2.
Definition.
Let formula_3 be a natural number. We define the digit sum for base formula_4 formula_5 to be the following:
where formula_7 is the number of digits in the number in base formula_8, and
is the value of each digit of the number. 
For example, in base 10, the digit sum of 84001 is formula_10.
For any two bases formula_11 and for sufficiently large natural numbers formula_3,
The sum of the base 10 digits of the integers 0, 1, 2, ... is given by in the On-Line Encyclopedia of Integer Sequences. 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.
Applications.
The concept of a decimal digit sum is closely related to, but not the same as, the digital root, which is the result of repeatedly applying the digit sum operation until the remaining value is only a single digit. The digital root of any non-zero integer will be a number in the range 1 to 9, whereas the digit sum can take any value. Digit sums and digital roots can be used for quick divisibility tests: a natural number is divisible by 3 or 9 if and only if its digit sum (or digital root) is divisible by 3 or 9, respectively. For divisibility by 9, this test is called the rule of nines and is the basis of the casting out nines technique for checking calculations.
Digit sums are also a common ingredient in checksum algorithms to check the arithmetic operations of early computers. Earlier, in an era of hand calculation, suggested using sums of 50 digits taken from mathematical tables of logarithms as a form of random number generation; if one assumes that each digit is random, then by the central limit theorem, these digit sums will have a random distribution closely approximating a Gaussian distribution.
The digit sum of the binary representation of a number is known as its Hamming weight or population count; algorithms for performing this operation have been studied, and it has been included as a built-in operation in some computer architectures and some programming languages. These operations are used in computing applications including cryptography, coding theory, and computer chess.
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.