TableOfPoliteNumberRepresentationsFor1N101 2009-01-20 20:58:51 2009-01-20 20:58:51 Example polite number % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here 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 \PMlinkname{number of (nondistinct) prime factors function}{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1}^n i$$ are also representable as $$\sum_{i = 0}^n i.$$ For sums with more than three addends, the middle addends have been replaced by three dots. \begin{tabular}{|r|l|l|l|l|l|r|r|r|} $n$ & Rep 1 & Rep 2 & Rep 3 & Rep 4 & Rep 5 & $N_p (n)$ & $\Omega(n)$ & $\Omega(n) - N_p (n)$ \\ 2 & & & & & & 0 & 1 & 1 \\ 3 & 1 + 2 & & & & & 1 & 1 & 0 \\ 4 & & & & & & 0 & 2 & 2 \\ 5 & 2 + 3 & & & & & 1 & 1 & 0 \\ 6 & 1 + 2 + 3 & & & & & 1 & 2 & 1 \\ 7 & 3 + 4 & & & & & 1 & 1 & 0 \\ 8 & & & & & & 0 & 3 & 3 \\ 9 & 4 + 5 & 2 + 3 + 4 & & & & 2 & 2 & 0 \\ 10 & 1 ... 4 & & & & & 1 & 2 & 1 \\ 11 & 5 + 6 & & & & & 1 & 1 & 0 \\ 12 & 3 + 4 + 5 & & & & & 1 & 3 & 2 \\ 13 & 6 + 7 & & & & & 1 & 1 & 0 \\ 14 & 2 ... 5 & & & & & 1 & 2 & 1 \\ 15 & 7 + 8 & 4 + 5 + 6 & 1 ... 5 & & & 3 & 2 & 1 \\ 16 & & & & & & 0 & 4 & 4 \\ 17 & 8 + 9 & & & & & 1 & 1 & 0 \\ 18 & 5 + 6 + 7 & 3 ... 6 & & & & 2 & 3 & 1 \\ 19 & 9 + 10 & & & & & 1 & 1 & 0 \\ 20 & 2 ... 6 & & & & & 1 & 3 & 2 \\ 21 & 10 + 11 & 6 + 7 + 8 & 1 ... 6 & & & 3 & 2 & $-1$ \\ 22 & 4 ... 7 & & & & & 1 & 2 & 1 \\ 23 & 11 + 12 & & & & & 1 & 1 & 0 \\ 24 & 7 + 8 + 9 & & & & & 1 & 4 & 3 \\ 25 & 12 + 13 & 3 ... 7 & & & & 2 & 2 & 0 \\ 26 & 5 ... 8 & & & & & 1 & 2 & 1 \\ 27 & 13 + 14 & 8 + 9 + 10 & 2 ... 7 & & & 3 & 3 & 0 \\ 28 & 1 ... 7 & & & & & 1 & 3 & 2 \\ 29 & 14 + 15 & & & & & 1 & 1 & 0 \\ 30 & 9 + 10 + 11 & 6 ... 9 & 4 ... 8 & & & 3 & 3 & 0 \\ 31 & 15 + 16 & & & & & 1 & 1 & 0 \\ 32 & & & & & & 0 & 5 & 5 \\ 33 & 16 + 17 & 10 + 11 + 12 & 3 ... 8 & & & 3 & 2 & $-1$ \\ 34 & 7 ... 10 & & & & & 1 & 2 & 1 \\ 35 & 17 + 18 & 5 ... 9 & 2 ... 8 & & & 3 & 2 & $-1$ \\ 36 & 11 + 12 + 13 & 1 ... 8 & & & & 2 & 4 & $-2$ \\ 37 & 18 + 19 & & & & & 1 & 1 & 0 \\ 38 & 8 ... 11 & & & & & 1 & 2 & 1 \\ 39 & 19 + 20 & 12 + 13 + 14 & 4 ... 9 & & & 3 & 2 & $-1$ \\ 40 & 6 ... 10 & & & & & 1 & 4 & 3 \\ 41 & 20 + 21 & & & & & 1 & 1 & 0 \\ 42 & 13 + 14 + 15 & 9 ... 12 & 3 ... 9 & & & 3 & 3 & 0 \\ 43 & 21 + 22 & & & & & 1 & 1 & 0 \\ 44 & 7 ... 11 & 2 ... 9 & & & & 2 & 3 & 1 \\ 45 & 22 + 23 & 14 + 15 + 16 & 5 ... 10 & 1 ... 9 & & 4 & 3 & $-1$ \\ 46 & 10 ... 13 & & & & & 1 & 2 & 1 \\ 47 & 23 + 24 & & & & & 1 & 1 & 0 \\ 48 & 15 + 16 + 17 & & & & & 1 & 5 & 4 \\ 49 & 24 + 25 & 4 ... 10 & & & & 2 & 2 & 0 \\ 50 & 11 ... 14 & 8 ... 12 & & & & 2 & 3 & 1 \\ 51 & 25 + 26 & 16 + 17 + 18 & 6 ... 11 & & & 3 & 2 & $-1$ \\ 52 & 3 ... 10 & & & & & 1 & 3 & 2 \\ 53 & 26 + 27 & & & & & 1 & 1 & 0 \\ 54 & 17 + 18 + 19 & 12 ... 15 & 2 ... 10 & & & 3 & 4 & 1 \\ 55 & 27 + 28 & 9 ... 13 & 1 ... 10 & & & 3 & 2 & $-1$ \\ 56 & 5 ... 11 & & & & & 1 & 4 & 3 \\ 57 & 28 + 29 & 18 + 19 + 20 & 7 ... 12 & & & 3 & 2 & $-1$ \\ 58 & 13 ... 16 & & & & & 1 & 2 & 1 \\ 59 & 29 + 30 & & & & & 1 & 1 & 0 \\ 60 & 19 + 20 + 21 & 10 ... 14 & 4 ... 11 & & & 3 & 4 & 1 \\ 61 & 30 + 31 & & & & & 1 & 1 & 0 \\ 62 & 14 ... 17 & & & & & 1 & 2 & 1 \\ 63 & 31 + 32 & 20 + 21 + 22 & 8 ... 13 & 6 ... 12 & 3 ... 11 & 5 & 3 & $-2$ \\ 64 & & & & & & 0 & 6 & 6 \\ 65 & 32 + 33 & 11 ... 15 & 2 ... 11 & & & 3 & 2 & $-1$ \\ 66 & 21 + 22 + 23 & 15 ... 18 & 1 ... 11 & & & 3 & 3 & 0 \\ 67 & 33 + 34 & & & & & 1 & 1 & 0 \\ 68 & 5 ... 12 & & & & & 1 & 3 & 2 \\ 69 & 34 + 35 & 22 + 23 + 24 & 9 ... 14 & & & 3 & 2 & $-1$ \\ 70 & 16 ... 19 & 12 ... 16 & 7 ... 13 & & & 3 & 3 & 0 \\ 71 & 35 + 36 & & & & & 1 & 1 & 0 \\ 72 & 23 + 24 + 25 & 4 ... 12 & & & & 2 & 5 & 3 \\ 73 & 36 + 37 & & & & & 1 & 1 & 0 \\ 74 & 17 ... 20 & & & & & 1 & 2 & 1 \\ 75 & 37 + 38 & 24 + 25 + 26 & 13 ... 17 & 10 ... 15 & 3 ... 12 & 5 & 3 & $-2$ \\ 76 & 6 ... 13 & & & & & 1 & 3 & 2 \\ 77 & 38 + 39 & 8 ... 14 & 2 ... 12 & & & 3 & 2 & $-1$ \\ 78 & 25 + 26 + 27 & 18 ... 21 & 1 ... 12 & & & 3 & 3 & 0 \\ 79 & 39 + 40 & & & & & 1 & 1 & 0 \\ 80 & 14 ... 18 & & & & & 1 & 6 & 5 \\ 81 & 40 + 41 & 26 + 27 + 28 & 11 ... 16 & 5 ... 13 & & 4 & 4 & 0 \\ 82 & 19 ... 22 & & & & & 1 & 2 & 1 \\ 83 & 41 + 42 & & & & & 1 & 1 & 0 \\ 84 & 27 + 28 + 29 & 9 ... 15 & 7 ... 14 & & & 3 & 4 & 1 \\ 85 & 42 + 43 & 15 ... 19 & 4 ... 13 & & & 3 & 2 & $-1$ \\ 86 & 20 ... 23 & & & & & 1 & 2 & 1 \\ 87 & 43 + 44 & 28 + 29 + 30 & 12 ... 17 & & & 3 & 2 & $-1$ \\ 88 & 3 ... 13 & & & & & 1 & 4 & 3 \\ 89 & 44 + 45 & & & & & 1 & 1 & 0 \\ 90 & 29 + 30 + 31 & 21 ... 24 & 16 ... 20 & 6 ... 14 & 2 ... 13 & 5 & 4 & $-1$ \\ 91 & 45 + 46 & 10 ... 16 & 1 ... 13 & & & 3 & 2 & $-1$ \\ 92 & 8 ... 15 & & & & & 1 & 3 & 2 \\ 93 & 46 + 47 & 30 + 31 + 32 & 13 ... 18 & & & 3 & 2 & $-1$ \\ 94 & 22 ... 25 & & & & & 1 & 2 & 1 \\ 95 & 47 + 48 & 17 ... 21 & 5 ... 14 & & & 3 & 2 & $-1$ \\ 96 & 31 + 32 + 33 & & & & & 1 & 6 & 5 \\ 97 & 48 + 49 & & & & & 1 & 1 & 0 \\ 98 & 23 ... 26 & 11 ... 17 & & & & 2 & 3 & 1 \\ 99 & 49 + 50 & 32 + 33 + 34 & 14 ... 19 & 7 ... 15 & 4 ... 14 & 5 & 3 & $-2$ \\ 100 & 18 ... 22 & 9 ... 16 & & & & 2 & 3 & 1 \\ \end{tabular}