# table of polite number representations for $1

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) $\Omega(n)$ 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 $T_{n}$, which in addition to being representable as

 $\sum_{i=1}^{n}i$

are also representable as

 $\sum_{i=0}^{n}i.$

For sums with more than three addends, the middle addends have been replaced by three dots.

$n$ 2 3 4 5 6 7 8 Rep 1 Rep 2 Rep 3 Rep 4 Rep 5 $N_{p}(n)$ $\Omega(n)$ $\Omega(n)-N_{p}(n)$ 0 1 1 1 + 2 1 1 0 0 2 2 2 + 3 1 1 0 1 + 2 + 3 1 2 1 3 + 4 1 1 0 0 3 3 4 + 5 2 + 3 + 4 2 2 0 1 … 4 1 2 1 5 + 6 1 1 0 3 + 4 + 5 1 3 2 6 + 7 1 1 0 2 … 5 1 2 1 7 + 8 4 + 5 + 6 1 … 5 3 2 1 0 4 4 8 + 9 1 1 0 5 + 6 + 7 3 … 6 2 3 1 9 + 10 1 1 0 2 … 6 1 3 2 10 + 11 6 + 7 + 8 1 … 6 3 2 $-1$ 4 … 7 1 2 1 11 + 12 1 1 0 7 + 8 + 9 1 4 3 12 + 13 3 … 7 2 2 0 5 … 8 1 2 1 13 + 14 8 + 9 + 10 2 … 7 3 3 0 1 … 7 1 3 2 14 + 15 1 1 0 9 + 10 + 11 6 … 9 4 … 8 3 3 0 15 + 16 1 1 0 0 5 5 16 + 17 10 + 11 + 12 3 … 8 3 2 $-1$ 7 … 10 1 2 1 17 + 18 5 … 9 2 … 8 3 2 $-1$ 11 + 12 + 13 1 … 8 2 4 $-2$ 18 + 19 1 1 0 8 … 11 1 2 1 19 + 20 12 + 13 + 14 4 … 9 3 2 $-1$ 6 … 10 1 4 3 20 + 21 1 1 0 13 + 14 + 15 9 … 12 3 … 9 3 3 0 21 + 22 1 1 0 7 … 11 2 … 9 2 3 1 22 + 23 14 + 15 + 16 5 … 10 1 … 9 4 3 $-1$ 10 … 13 1 2 1 23 + 24 1 1 0 15 + 16 + 17 1 5 4 24 + 25 4 … 10 2 2 0 11 … 14 8 … 12 2 3 1 25 + 26 16 + 17 + 18 6 … 11 3 2 $-1$ 3 … 10 1 3 2 26 + 27 1 1 0 17 + 18 + 19 12 … 15 2 … 10 3 4 1 27 + 28 9 … 13 1 … 10 3 2 $-1$ 5 … 11 1 4 3 28 + 29 18 + 19 + 20 7 … 12 3 2 $-1$ 13 … 16 1 2 1 29 + 30 1 1 0 19 + 20 + 21 10 … 14 4 … 11 3 4 1 30 + 31 1 1 0 14 … 17 1 2 1 31 + 32 20 + 21 + 22 8 … 13 6 … 12 3 … 11 5 3 $-2$ 0 6 6 32 + 33 11 … 15 2 … 11 3 2 $-1$ 21 + 22 + 23 15 … 18 1 … 11 3 3 0 33 + 34 1 1 0 5 … 12 1 3 2 34 + 35 22 + 23 + 24 9 … 14 3 2 $-1$ 16 … 19 12 … 16 7 … 13 3 3 0 35 + 36 1 1 0 23 + 24 + 25 4 … 12 2 5 3 36 + 37 1 1 0 17 … 20 1 2 1 37 + 38 24 + 25 + 26 13 … 17 10 … 15 3 … 12 5 3 $-2$ 6 … 13 1 3 2 38 + 39 8 … 14 2 … 12 3 2 $-1$ 25 + 26 + 27 18 … 21 1 … 12 3 3 0 39 + 40 1 1 0 14 … 18 1 6 5 40 + 41 26 + 27 + 28 11 … 16 5 … 13 4 4 0 19 … 22 1 2 1 41 + 42 1 1 0 27 + 28 + 29 9 … 15 7 … 14 3 4 1 42 + 43 15 … 19 4 … 13 3 2 $-1$ 20 … 23 1 2 1 43 + 44 28 + 29 + 30 12 … 17 3 2 $-1$ 3 … 13 1 4 3 44 + 45 1 1 0 29 + 30 + 31 21 … 24 16 … 20 6 … 14 2 … 13 5 4 $-1$ 45 + 46 10 … 16 1 … 13 3 2 $-1$ 8 … 15 1 3 2 46 + 47 30 + 31 + 32 13 … 18 3 2 $-1$ 22 … 25 1 2 1 47 + 48 17 … 21 5 … 14 3 2 $-1$ 31 + 32 + 33 1 6 5 48 + 49 1 1 0 23 … 26 11 … 17 2 3 1 49 + 50 32 + 33 + 34 14 … 19 7 … 15 4 … 14 5 3 $-2$ 18 … 22 9 … 16 2 3 1
Title table of polite number representations for $1 TableOfPoliteNumberRepresentationsFor1N101 2013-03-22 18:45:12 2013-03-22 18:45:12 PrimeFan (13766) PrimeFan (13766) 4 PrimeFan (13766) Example msc 11A25