How Do I Interpret This Notation?

[straight_stewie]

For fun, I'm trying to calculate the sum of all non abundant positive numbers. I have successfully done this by calculating factors, then summing factors and determining if a number is abundant or perfect, then summing the ones that aren't.

 

However, that runs a little slow. I was searching for more information and I found this gem of a Wolfram page: http://mathworld.wolfram.com/AbundantNumber.html

 

The page gives a hint that would make this easy to accomplish with sets:
 

Quote

Every positive integer n with (mod_n)60 is abundant.

I am familiar with congruence, but I can't figure out how to read this notation, and a lengthy google search only turned up results about the "%" operator, and information about congruence.

 

Does anyone know what this quote is saying?

[bcguru9384]

on youtube search

qbasic

there is a series of videos that may help

[Mira Yurizaki]

https://en.wikipedia.org/wiki/Modular_arithmetic#Congruence_relation

 

I think this is your answer.

[VicBar]

As I understand it. Abundant numbers are those where: The divisor function of that number minus that number is greater than that number. That being said, the divisor function is the sum of all of the powers of that numbers divisors.... Wheww... Interesting!

 

So mod, modulus, %, modulo, is an operation that returns the remainder of a division. It is used quite a bit in programming typically when learning to convert to different bases, because you divide by the base and use the remainder as the digit. 

 

That being said I'm not sure what the quote means with (mod_n)60 <- this part i don't understand.

 

[Mira Yurizaki]

So after digging around, I think what it's saying is every positive integer n is an abundant number if it's divisible by 60, including itself. I've seen a few sequences where 60, 120, 180, 240, etc. is in there (well they don't seem to go past 240 so...)

[VicBar]

Btw what language are you programming in?

[straight_stewie]

3 hours ago, VicBar said:

That being said I'm not sure what the quote means with (mod_n)60 <- this part i don't understand.

That's the part I'm hung on too. 
 

3 hours ago, M.Yurizaki said:

I think this is your answer.

I'd never seen this notation. 

So that means that this: 

3 hours ago, straight_stewie said:

Every positive integer n with (mod_n)60 is abundant.

Means:
Every positive integer n is abundant if n - 60 is a multiple of n?

EDIT:: That can't be right.

[VicBar]

What Languague tho? Maybe you can post it and we can help you optimize it.!

[Mira Yurizaki]

1 hour ago, VicBar said:

What Languague tho? Maybe you can post it and we can help you optimize it.!

I don't believe OP is asking about a programming language (unless you count whatever Wolfram Alpha uses)

 

2 hours ago, straight_stewie said:

Means:

Every positive integer n is abundant if n - 60 is a multiple of n?

EDIT:: That can't be right.

No, I think what they're saying is every positive integer n is an abundant number if it's a multiple of 60.

[straight_stewie]

1 hour ago, M.Yurizaki said:

No, I think what they're saying is every positive integer n is an abundant number if it's a multiple of 60.

I think you're right. Which means this fact is useless for my needs.

[VicBar]

14 hours ago, M.Yurizaki said:

No, I think what they're saying is every positive integer n is an abundant number if it's a multiple of 60.

This isn't quite so. If you look at the list of abundant numbers they do not start at 60. https://en.wikipedia.org/wiki/Abundant_number

What is True is that every multiple of 60 is an abundant number. But not the other way around.

And sadly yeah it doesn't help you code @straight_stewie...

14 hours ago, M.Yurizaki said:

I don't believe OP is asking about a programming language (unless you count whatever Wolfram Alpha uses

His post is because he finds his program a tad slow and was looking for a more optimal solution, such as using the mathematical function that defines abundant numbers.. seeing as we can't figure out quite how to interpret what he found on Wolfram Alpha, I was proposing we can optimize his code another way by looking at it.. Maybe the algorithm can be redesigned or the data structure he uses... That's why I ask what language he did it in. I don't know all programming languages.

 

[Mira Yurizaki]

10 minutes ago, VicBar said:

What is True is that every multiple of 60 is an abundant number.

That's what I said.

10 minutes ago, VicBar said:

His post is because he finds his program a tad slow and was looking for a more optimal solution, such as using the mathematical function that defines abundant numbers.. seeing as we can't figure out quite how to interpret what he found on Wolfram Alpha, I was proposing we can optimize his code another way by looking at it.. Maybe the algorithm can be redesigned or the data structure he uses... That's why I ask what language he did it in. I don't know all programming languages.

OP is only describing a design, not an implementation. The programming language is irrelevant (more or less).

[VicBar]

 

Quote

I have successfully done this by calculating factors, then summing factors and determining if a number is abundant or perfect, then summing the ones that aren't.

However, that runs a little slow.

@M.Yurizaki, he is definitely talking about some implementation he has done. And this is Programming, so I thought it would offer to help with whatever code he has been running.

Quote

No, I think what they're saying is every positive integer n is an abundant number if it's a multiple of 60

What this says implies is that no multiple of 60 is an abundant number. 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60 .... are abundant numbers.

Sorry for being so anal about this but I'm actually in Descrete Math class right now

[vorticalbox]

Quote

A number such that the sum of all its divisors (except itself) is greater than the number itself. Thus 12 is an abundant number, because its divisors, 1, 2, 3, 4, 6 = 16, which is greater than 12.

You basically add up all the divisions and if that is more than it is abundant. 

 

20 hours ago, straight_stewie said:

For fun, I'm trying to calculate the sum of all non abundant positive numbers. I have successfully done this by calculating factors, then summing factors and determining if a number is abundant or perfect, then summing the ones that aren't.

 

However, that runs a little slow. I was searching for more information and I found this gem of a Wolfram page: http://mathworld.wolfram.com/AbundantNumber.html

 

The page gives a hint that would make this easy to accomplish with sets:
 

I am familiar with congruence, but I can't figure out how to read this notation, and a lengthy google search only turned up results about the "%" operator, and information about congruence.

 

Does anyone know what this quote is saying?

Seeing as you have done that task and looking to optimise it maybe you should post your solution. 

[Mira Yurizaki]

2 hours ago, VicBar said:

What this says implies is that no multiple of 60 is an abundant number. 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60 .... are abundant numbers.

Sorry for being so anal about this but I'm actually in Descrete Math class right now

If I said "every positive integer n is an abundant number only if it's a multiple of 60", then what you're saying would be correct.

 

I mean, I could word it as "every positive integer n if it's a multiple of 60 is an abundant number."

 

EDIT: I do realize that also implies an exclusive set, but eh, English is rampant with confusing syntax.

[straight_stewie]

1 hour ago, vorticalbox said:

Seeing as you have done that task and looking to optimise it maybe you should post your solution. 

I'll post it in a while. I'm in the middle of optimizing the function that determines whether or not a number C can be written as the sum of two numbers in an array A, such that      A_i + A_j = C. Initially I had implemented the solution the naive way, try every possible sum.
 

[Blade of Grass]

5 hours ago, straight_stewie said:

I'll post it in a while. I'm in the middle of optimizing the function that determines whether or not a number C can be written as the sum of two numbers in an array A, such that      A_i + A_j = C. Initially I had implemented the solution the naive way, try every possible sum.
 

Are your numbers sorted?

[SpaceNugget]

On 2017-03-06 at 0:49 PM, straight_stewie said:

For fun, I'm trying to calculate the sum of all non abundant positive numbers. I have successfully done this by calculating factors, then summing factors and determining if a number is abundant or perfect, then summing the ones that aren't.

 

However, that runs a little slow. I was searching for more information and I found this gem of a Wolfram page: http://mathworld.wolfram.com/AbundantNumber.html

 

The page gives a hint that would make this easy to accomplish with sets:
 

I am familiar with congruence, but I can't figure out how to read this notation, and a lengthy google search only turned up results about the "%" operator, and information about congruence.

 

Does anyone know what this quote is saying?

mod partitions the set of real numbers. In first year math we used that notation. the set [6, 9 15] would satisfy (mod_n)3