table of polite number representations for
There clearly are patterns to the number of ways to represent a positive integer as a sum of consecutive nonnegative integers. There is only one way to represent odd primes in this manner, whereas composite odd numbers tend to have more representations
.
To try to make the relationship between integer factorization and number of representations as a sum of consecutive integers, the following table, in addition to listing the different sums and tallying them, also gives the value of the number of (nondistinct) prime factors
function (http://planetmath.org/NumberOfNondistinctPrimeFactorsFunction) and the difference
between the two. But to avoid needless repetition, the sums given are only of positive numbers; the only cases this makes a difference is for the triangular numbers
, which in addition to being representable as
are also representable as
For sums with more than three addends, the middle addends have been replaced by three dots.