The politeness of a positive number is defined as the number of ways it can be expressed as the sum of consecutive integers. For every x, the politeness of x equals the number of odd divisors of x that are greater than one. The politeness of the numbers 1, 2, 3, ... is
- 0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 3, 0, 1, 2, 1, 1, 3, ... (sequence A069283 in OEIS).
For instance, the politeness of 9 is 2 because it has two odd divisors, 3 and itself, and two polite representations
- 9 = 2 + 3 + 4 = 4 + 5;
the politeness of 15 is 3 because it has three odd divisors, 3, 5, and 15, and (as is familiar to cribbage players) three polite representations
- 15 = 4 + 5 + 6 = 1 + 2 + 3 + 4 + 5 = 7 + 8.
Read more about this topic: Polite Number
Famous quotes containing the word politeness:
“People are less self-conscious in the intimacy of family life and during the anxiety of a great sorrow. The dazzling varnish of an extreme politeness is then less in evidence, and the true qualities of the heart regain their proper proportions.”
—Stendhal [Marie Henri Beyle] (1783–1842)
“The highest perfection of politeness is only a beautiful edifice, built, from the base to the dome, of ungraceful and gilded forms of charitable and unselfish lying.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)
“Nobody thanks a witty man for politeness when he accommodates himself to a society in which it is not polite to display wit.”
—Friedrich Nietzsche (1844–1900)