Problem 2595. Polite numbers. Politeness.

Created by Jan Orwat

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A polite number is an integer that sums of two or more consecutive positive integers. Politeness of a positive integer is a number of nontrivial ways to write n as a sum of two or more consecutive positive integers.

For example 9 = 4+5 = 2+3+4 and politeness of 9 is 2.

Given N return politeness of N.

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Last Solution submitted on Nov 03, 2024

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Daniel Pereira on 27 Jan 2016

There are some flaws when checking the solutions:

I think 15 has 4 combinations of sum of consecutive INTEGERS (as stated in the problem):
15 = 7+8 = 4+5+6 = 1+2+3+4+5 = 0+1+2+3+4+5; % 0 is integer.

or 1025 (you say 5, I say 9):
1025 = 1022+1023 = sum(203:207) = sum(98:107) = sum(29:53) = sum(5:45) = sum(-4:45) = sum(-28:53) = sum(-97:107) = sum(-202:207); % negative numbers are integers too.

Jan Orwat on 28 Jan 2016

I can see the flaw in the description now, I've missed repeating "positive", Thanks. Btw considering your interpretation 15 has 7 combinations: sum(-14:15),sum(-6:8),sum(-3:6),sum(0:5),sum(1:5),sum(4:6),sum(7:8). In this way 1025 should have 11 and conversion between our interpretations is "yours=2*mine+1"; to any of mine solutions of the form m:n, you can add "-m+1:n", the last thing is to add "-input+1:input"

John D'Errico on 21 Aug 2016

An interesting problem, enough so that I chose to solve it in three essentially different ways. As always, there are various ways to solve any problem. The first two ways were essentially constructive, so counting the set of solutions for any N. The last used a formulaic approach.

Rafael S.T. Vieira on 11 Aug 2020

Politeness is an integer sequence defined at https://oeis.org/A069283.

goc3 on 13 Jan 2021

@Dyuman Joshi: I do not know why that error occurs. I do know that it essentially means that the user needs to wait and re-submit their solution at a later time, sometimes the next day.

By the way, it's best to not post solutions (or solution attempts) in comments. Questions or comments specific to a solution can be posted in a comment tied to said solution or solution attempt.

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Problem 2595. Polite numbers. Politeness.

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Problem 2595. Polite numbers. Politeness.

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