Four conditions

Number Theory Level 4

What is the smallest odd squarefree abundant number ?

Definition :

An abundant number is a number for which the sum of its proper divisors is greater than the number itself. For example, the proper divisors of 12 are 1, 2, 3, 4, 6, and the sum of these proper divisor exceed the number 12, thus 12 is an abundant number.


The answer is 15015.

Jake Lai by Jake Lai , 21, Singapore

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Otto Bretscher
Mar 28, 2016

Charming problem!

σ ( 3 × 5 × 7 × 11 × 13 ) = 32256 > 2 × 15015 \sigma(3\times5\times7\times11\times13)=32256>2\times15015 . But if n = p 1 × . . . × p m n=p_1\times...\times p_m for four or fewer distinct odd prime factors, then σ ( n ) n = ( 1 + 1 p 1 ) × . . . × ( 1 + 1 p m ) ≤ ( 1 + 1 3 ) ( 1 + 1 5 ) ( 1 + 1 7 ) ( 1 + 1 11 ) < 2 \frac{\sigma(n)}{n}=(1+\frac{1}{p_1})\times...\times(1+\frac{1}{p_m})\leq(1+\frac{1}{3})(1+\frac{1}{5})(1+\frac{1}{7})(1+\frac{1}{11})<2 .

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Thank you! Appreciate this solution, succinct and clear as always :)

Jake Lai - 5 years, 2 months ago

It is 32256 :)

Tran Hieu - 5 years, 2 months ago

Log in to reply

true...corrected...thanks!

Otto Bretscher - 5 years, 2 months ago

I thought it had to be the sum of the divisors not the product

Greg Grapsas - 3 years, 4 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...