Gnumbers

Gnumbers are positive integers that can be expressed as the sum of at least two consecutive positive integers. For example, 3 equals 1 plus 2, 6 equals 1 plus 2 plus 3, 9 equals 4 plus 5, 12 equals 3 plus 4 plus 5, and 15 can be written as 1+2+3+4+5 or as 7+8. The concept is used in recreational and introductory number theory to illustrate representations of numbers as sums of consecutive terms; powers of two are not Gnumbers because they cannot be formed as such a sum.

Characterization: A classic result states that a positive integer n can be written as a sum of two or more consecutive positive integers if and only if n is not a power of two. If n has an odd divisor greater than 1, then n can be expressed as a sum of that many consecutive integers centered around n divided by the divisor. Conversely, if n is a power of two, no such representation exists because any sum of consecutive positive integers has an odd length or an odd divisor structure that cannot produce a pure power of two.

Representations: The number of distinct representations of n as a sum of consecutive positive integers equals the number of odd divisors of n. Consequently, non-power-of-two numbers have at least one such representation, while powers of two have none. This connection to the divisor structure of n provides a tidy way to determine the existence and quantity of representations.

See also: consecutive integers, divisor functions, perfect numbers.