## A sequence of adding the product of their prime factors

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### A sequence of adding the product of their prime factors

I was curious to what would happen when you take a number, split it to its prime factors, add them up and then to repeat that sequence till you eventually get to a prime number. I was wondering what is the effect of this and what would happen the further you go up, I went up to one hundred and I think the higher you go the less likely 5 will be in the next number, as there are more and more primes. 4 is the only number which ends up to 4 and 5 is the first number the primes can go to, as 2 and 3 are to small because they can't be added with different primes. I know this may sound really confusing but I hope you understand. I'm just wondering if you have any information on this sequence? And is the pattern random or will you be able to know when the next number ends to 5 and the next ends to 13? This is what I've got (the first term being number 1):
1, 2, 3, 4, 5, 5, 7, 5, 5, 7, 11, 7, 13, 5, 5, 5, 17, 11, 19, 5, 7, 13, 23, 5, 7, 5, 7, 11, 29, 7, 31, 7, 5, 19, 7, 7

I think this sequence is interesting but I have no idea whether it's random or more numbers appear more than others further up you go. Since primes are rather random themselves does that make this sequence random too? What do you think?
offscott

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Joined: Sat Apr 21, 2012 9:57 pm UTC

### Re: A sequence of adding the product of their prime factors

I may be misunderstanding the sequence you've described, here's what it looks like you're saying:

I can start with any number; I'll choose 100 here. So that's the first term in the sequence.
100 factors in 2*2*5*5, and 2+2+5+5 = 14. That's the second term in the sequence.
14 factors into 2*7, and 2+7 = 9. That's the third term.
9 factors into 3*3, and 3+3 = 6.
6 = 2*3, 2+3 = 5, which is prime and thus does not factor any further.
So the sequence would be 100, 14, 9, 6, 5. Starting with a different number will of course give a different sequence. But that doesn't at all match the sequence you gave, so at least one of us is misunderstanding something.
Last edited by Meteoric on Sun Apr 22, 2012 6:17 am UTC, edited 2 times in total.
No, even in theory, you cannot build a rocket more massive than the visible universe.
Meteoric

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Joined: Wed Nov 23, 2011 4:43 am UTC

### Re: A sequence of adding the product of their prime factors

The "sum of prime factors" fucntion or sopfr is defined here. http://mathworld.wolfram.com/SumofPrimeFactors.html you want to look at the fourth paragrph where they iterate to a fixed point. The only difference between yours and theirs is that their sum of the factors of one (other than one or itself) is zero, Else they appear to use the same convention.

http://oeis.org/A029908/b029908.txt has a list of the first 1000.
Yesila

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Joined: Sun Dec 16, 2007 11:38 am UTC

### Re: A sequence of adding the product of their prime factors

Meteoric wrote: But that doesn't at all match the sequence you gave, so at least one of us is misunderstanding something.

The op's sequence is listing where the number 1 goes, then where 2, goes, then where 3 goes.... so, for example, the 12th number in their list (since 12=2*2*3) is 2+2+3=7.
Yesila

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Joined: Sun Dec 16, 2007 11:38 am UTC

### Re: A sequence of adding the product of their prime factors

Ah, I get it.
No, even in theory, you cannot build a rocket more massive than the visible universe.
Meteoric

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Joined: Wed Nov 23, 2011 4:43 am UTC

### Re: A sequence of adding the product of their prime factors

Thank you ^^ I gave it a good read and it was quite interesting.
offscott

Posts: 4
Joined: Sat Apr 21, 2012 9:57 pm UTC