My Math Forum  

Go Back   My Math Forum > College Math Forum > Number Theory

Number Theory Number Theory Math Forum


Reply
 
LinkBack Thread Tools Display Modes
May 7th, 2016, 06:50 AM   #1
Newbie
 
Joined: May 2014
From: India

Posts: 7
Thanks: 0

Numbers x such that the sum of the divisors is a perfect square.

Hello I am reading "The Theory of Numbers, by Robert D. Carmichael" and stuck in an exercise problem,

Find numbers x such that the sum of the divisors of x is a perfect square.

I know sum of divisors of a x=$\displaystyle x=p_1^{{\alpha}_1}.p_2^{{\alpha}_2}...p_n^{{\alpha }_i}$ is

Sum of divisors =$\displaystyle =\prod{\frac{p_i^{{\alpha}_i+1}-1}{p_i-1}}$

But couldn't proceed further on how resolve the product in to X2

It will be helpful if someone supply some hints
teddybear is offline  
 
May 7th, 2016, 08:42 PM   #2
Senior Member
 
Joined: Feb 2012

Posts: 144
Thanks: 16

I think you are counting 1 several times.
mehoul is offline  
May 8th, 2016, 08:11 PM   #3
Global Moderator
 
greg1313's Avatar
 
Joined: Oct 2008
From: London, Ontario, Canada - The Forest City

Posts: 7,885
Thanks: 1088

Math Focus: Elementary mathematics and beyond
1, 3, 15, 105, 945 ... depending on how you factor them and using only proper divisors.
greg1313 is online now  
May 9th, 2016, 04:40 AM   #4
Global Moderator
 
Joined: Dec 2006

Posts: 19,992
Thanks: 1855

Also 12, as 1 + 2 + 3 + 4 + 6 = 16 = 4². However, the problem states "divisors", not "proper divisors".

As the problem doesn't ask for all numbers, it would suffice to provide 1, 3 and 22 (see here).
skipjack is offline  
May 9th, 2016, 01:43 PM   #5
Newbie
 
Joined: May 2014
From: India

Posts: 7
Thanks: 0

Thanks everyone; I have seen the OEIS series, skipjack.

I was wondering whether there is any general form for such numbers, as OEIS doesn't seem to mention it.

Last edited by skipjack; December 13th, 2016 at 11:53 PM.
teddybear is offline  
Reply

  My Math Forum > College Math Forum > Number Theory

Tags
divisors, numbers, perfect, square, sum, umbers



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Perfect square greg1313 Number Theory 5 November 28th, 2011 07:16 AM
Perfect square PRO Number Theory 6 August 3rd, 2011 06:38 PM
perfect square and perfect cube calligraphy Number Theory 4 February 10th, 2011 06:34 AM
NUMBER OF SQUARE-FULL DIVISORS Barbarel Number Theory 2 November 7th, 2009 02:28 PM
N square, sum of its divisors square momo Number Theory 2 September 18th, 2008 01:58 PM





Copyright © 2018 My Math Forum. All rights reserved.