# Sum of divisors function question

• Aug 29th 2010, 02:39 AM
Khonics89
Sum of divisors function question
Let p(i) denote the i-th prime number. (Thus (p1, p2, p3,....) = (2, 3, 5, ... ).)
Find the smallest positive integer k such that the product n = p1 p2 ... pk satisfies
s(n) > 3n. Is there any positive integer m < n satisfying s(m) > 3m?

s denotes the sum of divisors function.

Honestly I have 0 clue on how do I start a question like this...

It would be nice, if someone can lead me to the right way.

Thanks
• Aug 29th 2010, 03:02 AM
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Quote:

Originally Posted by Khonics89
Let p(i) denote the i-th prime number. (Thus (p1, p2, p3,....) = (2, 3, 5, ... ).)
Find the smallest positive integer k such that the product n = p1 p2 ... pk satisfies
s(n) > 3n. Is there any positive integer m < n satisfying s(m) > 3m?

s denotes the sum of divisors function.

Honestly I have 0 clue on how do I start a question like this...

It would be nice, if someone can lead me to the right way.

Thanks

By the way the sequence of n_i are called primorials.

I'm afraid I'm getting spoiled by my CAS and got the answer initially just by writing a loop. But reflecting a bit and using reference

Divisor Function -- from Wolfram MathWorld

note that s(ab) = s(a)s(b) and that s(p) = p+1 where p prime. So s(p_1*p_2*...*p_i) = (p_1+1)(p_2+1)...(p_i+1). So just build the sequence and stop when the condition is met.

The answer to the second question is yes, there are many such integers m.
• Aug 29th 2010, 11:30 PM
Khonics89
Undefined: Can you explain what the question is asking me?

I don't get the quesiton at all...
• Aug 29th 2010, 11:50 PM
undefined
Quote:

Originally Posted by Khonics89
Undefined: Can you explain what the question is asking me?

I don't get the quesiton at all...

Sum of divisors of n is what its name suggests.. so s(8) = 1 + 2 + 4 + 8.

2 with s(2),
2*3 with s(2*3),
2*3*5 with s(2*3*5),
etc.

What don't you understand?
• Aug 31st 2010, 01:18 AM
mel240
why are there many m such that s(m)>3m
Is it because m is defined as something different to n?
• Aug 31st 2010, 05:04 AM
undefined
Quote:

Originally Posted by mel240
why are there many m such that s(m)>3m
Is it because m is defined as something different to n?

For example consider these guys

Highly composite number - Wikipedia, the free encyclopedia