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For a non-square integer, n, every divisor, d, of n is paired with divisor n/d of n and () is even; for a square integer, one divisor (namely ) is not paired with a distinct divisor and () is odd. Similarly, the number σ 1 ( n ) {\displaystyle \sigma _{1}(n)} is odd if and only if n is a square or twice a square.
The tables below list all of the divisors of the numbers 1 to 1000. A divisor of an integer n is an integer m, for which n/m is again an integer (which is necessarily also a divisor of n). For example, 3 is a divisor of 21, since 21/7 = 3 (and therefore 7 is also a divisor of 21). If m is a divisor of n, then so is −m. The tables below only ...
A harmonic divisor number is a positive integer whose divisors have a harmonic mean that is an integer. The first five of these are 1, 6, 28, 140, and 270. It is not known whether any harmonic divisor numbers (besides 1) are odd, but there are no odd ones less than 10 24. The sum of the reciprocals of the divisors of a perfect number is 2.
In number theory, the aliquot sum s(n) of a positive integer n is the sum of all proper divisors of n, that is, all divisors of n other than n itself. That is, = |,. It can be used to characterize the prime numbers, perfect numbers, sociable numbers, deficient numbers, abundant numbers, and untouchable numbers, and to define the aliquot sequence of a number.
A classical example of this phenomenon [9] is given by the divisor summatory function, the summation function of d(n), the number of divisors of n: = = + + + + + + = An average order of an arithmetic function is some simpler or better-understood function which has the same summation function asymptotically, and ...
In number theory, a deficient number or defective number is a positive integer n for which the sum of divisors of n is less than 2n. Equivalently, it is a number for which the sum of proper divisors (or aliquot sum) is less than n. For example, the proper divisors of 8 are 1, 2, and 4, and their sum is less than 8, so 8 is deficient.
A simple example of the use of this formula is counting the number of reduced fractions 0 < a / b < 1, where a and b are coprime and b ≤ n. If we let f(n) be this number, then g(n) is the total number of fractions 0 < a / b < 1 with b ≤ n, where a and b are not necessarily coprime.
Other important values of the zeta function include () = /, the solution to the Basel problem, Apéry's constant (), proved by Roger Apéry to be an irrational number, and the "critical line" of complex numbers with real part , conjectured by the Riemann hypothesis to be the only values other than negative integers where the function can be zero.