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Determine whether $\sigma(n)<e^\gamma n \omega(n)$ for all $n$ not of the form $2^x$. In words (to define the symbols), the sum of the divisors of $n$ is less than the product of Euler's number to the power of the Euler-Mascheroni constant, $n$, and the number of distinct prime factors of $n$. Those familiar with the definition of $\dfrac{\sigma(n)}n$ as the "abundancy" of $n$ might notice that this equivalently says the ratio of $n$'s abundancy to its number of distinct prime factors is $<e^\gamma\approx 1.78107$.

It is known that $$\lim_{n \to \infty} \sup\sigma(n)=e^\gamma n \log\log n,$$ $$\omega(n) \sim \log\log n,$$ and Robin's theorem says $$\sigma(n)<e^\gamma n \log\log n$$ for all $n>5040$ is equivalent to the Riemann Hypothesis. That is mainly what interests me about the inequality. Clearly the inequality is stronger than Robin's inequality whenever $\omega(n)<\log\log n$, so for any number of distinct prime factors the inequality will be stronger for all $n>e^{\large{e^{\large{\omega(n)}}}}$. The fundamental theorem of arithmetic implies every integer $>1$ has at least one prime factor, but $e^{\large{e}}\approx 15.15426$, so $n<16 \implies \omega(n)>\log\log n$. It's easy to show that sufficiently large powers of two violate the inequality. Indeed every power of two greater than or equal to $16$ violates the inequality, but consider that for a prime $p$, $$\lim_{a \to \infty}\dfrac{\sigma(p^a)}{p^a}=\dfrac{p}{p-1},$$ so we can see that no power of any other prime will violate the inequality, since $3/(3-1)=(2+1)/2=\sigma(2)/2$, which we have already verified satisfies the inequality, and clearly $$\dfrac{n+1}{(n+1)-1}<\dfrac{n}{n-1}$$ for all $n>1$, since $$\dfrac{n}{n-1}-\dfrac{n+1}n=\dfrac{n^2-(n^2-1)}{n^2-n}=\dfrac{1}{n^2-n}.$$

Now we have seen what the absolute largest possible abundancy is for any given prime power, and the exponents on the primes chosen will not affect the number of distinct prime factors, so it is obvious that for any given number of distinct prime factors $\omega(n)=k$,

$$\dfrac{\sigma(n)}{n}<2\dfrac{3}2\dfrac{5}4...\dfrac{p_k}{p_k-1},$$

where $p_k$ is the $k$th prime. It is then obvious that if R.H.S. $<ke^\gamma$ for all $k>1$, then our central inequality holds for all non-powers of two, but otherwise there are certain to be counterexamples given sufficiently large exponents on the first $k$ primes.

Recall that Merten's $3$rd theorem tells us the R.H.S is asymptotically $e^\gamma \log p_k$ as $k$ approaches infinity, which brings us to the inequality

$$p_k<e^k,$$

but not rigorously, and this inequality easily follows from Bertrand's postulate. That is, we can show by induction that $p_k<2^k<e^k$.

This essentially demonstrates that the inequality will be true for all $n$ such that $\omega(n)\geq A$ for some constant $A$. The question is whether we can take $A=2$.

Can you show me how to prove the inequality from the beginning? I'm not sure if what I've done so far is correct, and I've skipped over some steps.

A side note: it appears that $\omega(n)\geq \log\log n$ consistently for smaller colossally abundant numbers, but I'm not sure how to prove this in general. I suppose a start would be to show that only finitely many colossally abundant numbers exist with any given number of distinct prime factors. I'm interested in this question because we should not expect to get a much better estimate than Robin's inequality for the colossally abundant numbers, since denoting $a_r$ the $r$th colossally abundant number, $$\lim_{r \to \infty}\sigma(a_r)=e^\gamma a_r\log\log a_r.$$

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You do not need all the machinery for this one. For a fixed $\omega(n),$ we need to do calculations for only the consecutive primes up to $\omega(n),$ the same collection of exponents on larger primes would shrink your ratio.

For $2^a,$ your ratio is less than $2$. For $2^a 3^b$ your ratio is less than $3/2.$ For $2^a 3^b 5^c$ your ratio is less than $5/4.$ And For $2^a 3^b 5^c 7^d$ your ratio is less than $35/32.$

Indeed, if I am using $k-1$ primes (with nonzero exponents) and increase to $k$ primes, the upper bound for the ratio is multiplied by $$ \left( \frac{k-1}{k} \right) \left( \frac{p_k}{p_k - 1} \right) $$ Since $p_k > k,$ this product is less than $1.$ As a result, adding a new prime shrinks the upper bound, always. We had already reached $3/2 < e^\gamma$ by the time we added the prime $3.$ So that's it, only the powers of $2$ work.

The business about using consecutive primes is the fact that $$ \left( \frac{p}{p - 1} \right) $$ decreases as $p$ increases. So, for any sinle prime the upper bound is that $\frac{p}{p - 1},$ so any choice of prime other than $2$ gives below $3/2.$ Any choice of two primes gives below $3/2.$ Any choice of three primes gives below $5/4.$

So, when I ask the computer for ratios above $35/32,$ I do get a mix of things; the thing that has been proved is that at most three primes will be used. These primes need not be consecutive, when searching in this manner.

        2 = 2                      1.5
           3 = 3        1.333333333333333
           4 = 2^2                     1.75
           5 = 5                      1.2
           7 = 7        1.142857142857143
           8 = 2^3                    1.875
           9 = 3^2        1.444444444444444
          12 = 2^2 * 3        1.166666666666667
          16 = 2^4                   1.9375
          24 = 2^3 * 3                     1.25
          25 = 5^2                     1.24
          27 = 3^3        1.481481481481481
          32 = 2^5                  1.96875
          36 = 2^2 * 3^2        1.263888888888889
          40 = 2^3 * 5                    1.125
          48 = 2^4 * 3        1.291666666666667
          49 = 7^2        1.163265306122449
          54 = 2 * 3^3        1.111111111111111
          64 = 2^6                 1.984375
          72 = 2^3 * 3^2        1.354166666666667
          80 = 2^4 * 5                   1.1625
          81 = 3^4        1.493827160493827
          96 = 2^5 * 3                   1.3125
         108 = 2^2 * 3^3        1.296296296296296
         112 = 2^4 * 7        1.107142857142857
         121 = 11^2        1.099173553719008
         125 = 5^3                    1.248
         128 = 2^7                1.9921875
         144 = 2^4 * 3^2        1.399305555555556
         160 = 2^5 * 5                  1.18125
         162 = 2 * 3^4         1.12037037037037
         192 = 2^6 * 3        1.322916666666667
         200 = 2^3 * 5^2                   1.1625
         216 = 2^3 * 3^3        1.388888888888889
         224 = 2^5 * 7                    1.125
         243 = 3^5        1.497942386831276
         256 = 2^8               1.99609375
         288 = 2^5 * 3^2                 1.421875
         320 = 2^6 * 5                 1.190625
         324 = 2^2 * 3^4        1.307098765432099
         343 = 7^3        1.166180758017493
         384 = 2^7 * 3                 1.328125
         400 = 2^4 * 5^2                  1.20125
         432 = 2^4 * 3^3        1.435185185185185
         448 = 2^6 * 7        1.133928571428571
         486 = 2 * 3^5        1.123456790123457
         512 = 2^9              1.998046875
         576 = 2^6 * 3^2        1.433159722222222
         625 = 5^4                   1.2496
         640 = 2^7 * 5                1.1953125
         648 = 2^3 * 3^4        1.400462962962963
         720 = 2^4 * 3^2 * 5        1.119444444444444
         729 = 3^6        1.499314128943759
         768 = 2^8 * 3        1.330729166666667
         784 = 2^4 * 7^2        1.126913265306122
         800 = 2^5 * 5^2                 1.220625
         864 = 2^5 * 3^3        1.458333333333333
         896 = 2^7 * 7        1.138392857142857
         972 = 2^2 * 3^5        1.310699588477366
        1000 = 2^3 * 5^3                     1.17
        1024 = 2^10             1.9990234375
        1080 = 2^3 * 3^3 * 5        1.111111111111111
        1152 = 2^7 * 3^2        1.438802083333333
        1280 = 2^8 * 5               1.19765625
        1296 = 2^4 * 3^4        1.447145061728395
        1331 = 11^3         1.09992486851991
        1440 = 2^5 * 3^2 * 5                   1.1375
        1458 = 2 * 3^6        1.124485596707819
        1536 = 2^9 * 3               1.33203125
        1568 = 2^5 * 7^2        1.145089285714286
        1600 = 2^6 * 5^2                1.2303125
        1728 = 2^6 * 3^3        1.469907407407407
        1792 = 2^8 * 7                 1.140625
        1800 = 2^3 * 3^2 * 5^2        1.119444444444444
        1944 = 2^3 * 3^5        1.404320987654321
        2000 = 2^4 * 5^3                    1.209
        2048 = 2^11            1.99951171875
        2160 = 2^4 * 3^3 * 5        1.148148148148148
        2187 = 3^7        1.499771376314586
        2304 = 2^8 * 3^2        1.441623263888889
        2401 = 7^4        1.166597251145356
        2560 = 2^9 * 5              1.198828125
        2592 = 2^5 * 3^4        1.470486111111111
        2880 = 2^6 * 3^2 * 5        1.146527777777778
        2916 = 2^2 * 3^6        1.311899862825789
        3072 = 2^10 * 3        1.332682291666667
        3125 = 5^5                  1.24992
        3136 = 2^6 * 7^2        1.154177295918367
        3200 = 2^7 * 5^2               1.23515625
        3240 = 2^3 * 3^4 * 5         1.12037037037037
        3456 = 2^7 * 3^3        1.475694444444444
        3584 = 2^9 * 7        1.141741071428571
        3600 = 2^4 * 3^2 * 5^2        1.156759259259259
        3888 = 2^4 * 3^5        1.451131687242798
        4000 = 2^5 * 5^3                   1.2285
        4096 = 2^12           1.999755859375
        4320 = 2^5 * 3^3 * 5        1.166666666666667
        4374 = 2 * 3^7         1.12482853223594
        4608 = 2^9 * 3^2        1.443033854166667
        5000 = 2^3 * 5^4                   1.1715
        5120 = 2^10 * 5             1.1994140625
        5184 = 2^6 * 3^4        1.482156635802469
        5400 = 2^3 * 3^3 * 5^2        1.148148148148148
        5488 = 2^4 * 7^3        1.129737609329446
        5760 = 2^7 * 3^2 * 5        1.151041666666667
        5832 = 2^3 * 3^6        1.405606995884774
        6048 = 2^5 * 3^3 * 7        1.111111111111111
        6144 = 2^11 * 3             1.3330078125
        6272 = 2^7 * 7^2        1.158721301020408
        6400 = 2^8 * 5^2              1.237578125
        6480 = 2^4 * 3^4 * 5        1.157716049382716
        6561 = 3^8        1.499923792104862
        6912 = 2^8 * 3^3        1.478587962962963
        7168 = 2^10 * 7        1.142299107142857
        7200 = 2^5 * 3^2 * 5^2        1.175416666666667
        7776 = 2^5 * 3^5        1.474537037037037
        8000 = 2^6 * 5^3                  1.23825
        8064 = 2^7 * 3^2 * 7        1.096230158730159
        8192 = 2^13          1.9998779296875
        8640 = 2^6 * 3^3 * 5        1.175925925925926
        8748 = 2^2 * 3^7        1.312299954275263
        9000 = 2^3 * 3^2 * 5^3        1.126666666666667
        9072 = 2^4 * 3^4 * 7        1.102586713697825
        9216 = 2^10 * 3^2        1.443739149305556
        9600 = 2^7 * 3 * 5^2        1.097916666666667
        9720 = 2^3 * 3^5 * 5        1.123456790123457
       10000 = 2^4 * 5^4                  1.21055
jagy@phobeusjunior:~$ 
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  • $\begingroup$ Thanks. I got most of this, but didn't realize this was all that was needed. I thought you might make an appearance. $\endgroup$ – Jaycob Coleman Oct 31 '13 at 17:32

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