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As usual, for a positive integer $n$, let $\sigma(n)$ denote the sum of all positive divisors of $n$ (including 1 and $n$).

What is the smallest ODD number such that $\sigma(n) \ge 3n$?

For comparison, the answers to some related questions are:

Smallest $n$ of any parity satisifying $\sigma(n) \ge 3n$ is $n=120=2^3\cdot3\cdot5$. We have $\sigma(120)=360$.

Smallest $n$ of any parity satisifying $\sigma(n) > 3n$ is $n=180=2^2\cdot3^2\cdot5$. We have $\sigma(180)=546$.

Smallest ODD $n$ satisifying $\sigma(n) \ge 2n$ is $n=945=3^3\cdot5\cdot7$. We have $\sigma(945)=1920$.

I suspect that the answer to my above question is $$ 1310112879075 = 3^5 \cdot 5^2 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 $$ but I can't quite prove it.

I can prove that the answer must be divisible by each of the primes from 3 to 23, and then I can use trial and error to compare various candidates (e.g. compare multiplication by 29 with multiplication by $3^3$, and such "playing around").

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I do not think that in general this kind of problem can be solved without some playing around. The fine structure of the order relations between powers of the primes is a mystery. – André Nicolas Feb 21 '12 at 17:27
up vote 5 down vote accepted

Nope, the smallest is $$ n=1018976683725=3^3 \cdot 5^2 \cdot 7^2 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29, $$ which has $\frac{\sigma(n)}{n}=\frac{34283520}{11350339}\approx 3.02048423399513$.

Just for the record, I found this by using your observation that any candidate must be a multiple of $m:=3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23$. Your answer is $9765m$, and by running a for loop over all smaller (odd) multiples, I came upon the (unique) smaller answer of $n=9135m$.

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Thank you. That is clearly smaller than my answer, and also satisfies sigma(n) > 3n. A minor quibble: I proved that any candidate is divisible by all primes up to 23, not necessarily 29. However, even if we redefine m, the same trick of looping through odd multiples of m still verifies that we've found the smallest. – idmercer Feb 21 '12 at 18:01
Oops, good catch. Edited for posterity. – Cam McLeman Feb 21 '12 at 18:09

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