For every odd $n\in\mathbb{N}$, is it true that $\sigma(n) < 2n$?

Question

Is the following proposition true?

Let $n \in \mathbb{N}$ be an odd number, then $\sigma(n) < 2n$ .

For $n=p_1^\alpha p_2^\beta$ it is true :

$$\sigma(n)=‎‎\left(\frac{p_1^{\alpha+1}-1}{P_1-1}\right)\left(‎\frac{p_2^{\beta+1}-1}{P_2-1}‎\right)‎‎ < 2p_1^{\alpha} p_2^{\beta} ‎\Longleftrightarrow‎ $$

$$2‎ < ‎‎\frac{p_1^{\alpha+1} p_2^{\beta+1}+p_1^{\alpha+1}+p_2^{\beta+1}-1}{p_1^{\alpha} p_2^{\beta}(p_1+p_2-1)}‎‎=‎\frac{{p_1}{p_2}}{p_1+p_2-1}+‎\frac{p_1^{\alpha+1}+p_2^{\beta+1}-1}{p_1^{\alpha} p_2^{\beta}(p_1+p_2-1)}$$

$$\Longrightarrow‎ 2< \frac{{p_1}{p_2}}{p_1+p_2-1}+‎\frac{p_1^{\alpha+1}+p_2^{\beta+1}-1}{p_1^{\alpha} p_2^{\beta}(p_1+p_2-1)}$$ This is true beacuse of $\frac{{p_1}{p_2}}{p_1+p_2-1} \geq 2$ and $\frac{p_1^{\alpha+1}+p_2^{\beta+1}-1}{p_1^{\alpha} p_2^{\beta}(p_1+p_2-1)} > 0$

Can somebody help me for $n=p_1^{\alpha_1} p_2^{\alpha_2}...p_m^{\alpha_m}$ ?

Thank you.

Answer 1

No, that's false. The numbers for which $\sigma(n) = 2n$ are called perfect, while $\sigma(n) > 2n$ are called abundant. It is currently unknown whether there are odd perfect numbers or not, while the smallest odd abundant number is $945$.

Even more: there are infinitely many odd abundant numbers (cfr. OEIS sequence A005231). This is because every integer multiple (so in particular every odd multiple) of an abundant number is again abundant.

Answer 2

For a counterexample, look at $n=945=3^3\cdot 5\cdot 7$. It turns out that $\sigma(n)=1920\gt 1890$. There are infinitely many other counterexamples. One family is obtained by multiplying $945$ by any odd number. There are other families.

Answer 3

An efficient way to produce infinitely many numbers with increasing values of $\frac{\sigma(n)}{n}$ while staying with odd numbers, let $n$ be the least common multiple of the consecutive odd numbers $1,3,5,7,9,11,\ldots.$ Not the product, the LCM. This method should produce ratios that slowly grow. Let me work up a table, give me a few minutes.

   odd      lcm     sigma  ratio
    1         1         1  1
    3         3         4  1.333333333333333
    5        15        24  1.6
    7       105       192  1.828571428571429
    9       315       624  1.980952380952381
   11      3465      7488  2.161038961038961
   13     45045    104832  2.327272727272727
   15     45045    104832  2.327272727272727
   17    765765   1886976  2.464171122994653
   19  14549535  37739520  2.593864339994371
   21  14549535  37739520  2.593864339994371
   23 334639305 905748480  2.706641050428909

It is not difficult to prove that the ratio stays the same if the new odd number is divisible by two or more different primes, but the ratio increases if the new odd number is a prime or prime power. In the long run, the exponent for some prime factor $p$ in the LCM is roughly proportional to $\frac{1}{\log p}.$

Another sequence, less efficient but easier to describe and to program, just takes the ordinary product of the consecutive prime numbers, beginning with $3.$ It is not even necessary to find the product or its sum of divisors, every time we use a new prime we just multiply the existing ratio by $(p+1)/p.$ This increases without bound, because the sum of prime reciprocals diverges. Note how the first three ratios in this list are the same as the first three in the earlier list.

 May 30, 2015 

    3  1.333333333333333
    5  1.6
    7  1.828571428571429
   11  1.994805194805195
   13  2.148251748251748
   17  2.27461949814891
   19  2.394336313840958
   23  2.498437892703608
   29  2.584590923486491
   31  2.66796482424412
   37  2.740071981656123
   41  2.806903005598956
   43  2.872179819682652
   47  2.93329002861207
   53  2.988635123491544
   59  3.039289956093095
   61  3.089114381602818
   67  3.13522056640286
   71  3.179378602549379
   73  3.222931734091151
   79  3.263728338320153
   83  3.303050366492685
   89  3.340163291958895
   97  3.374597965071873
  101  3.408009826112189
  103  3.441097300152113
  107  3.473257088004002
  109  3.505121831930644
  113  3.536140609204367
  127  3.563984236048496
  131  3.591190222583217
  137  3.617403289901343
  139  3.643427774001352
  149  3.667880309397335
  151  3.692170907472814
  157  3.715687919622322
  163  3.738483551031048
  167  3.760869680079138
  173  3.782608811177862
  179  3.803740703977738
  181  3.824755845988665
  191  3.844780745705883
  193  3.864701889466017
  197  3.884319665554677
  199  3.903838859853947
  211  3.922340465824818
  223  3.939929436523585
  227  3.957285953865098
  229  3.974566678554465
  233  3.991624904642682
  239  4.008326264076334
  241  4.024958323263372
  251  4.040994013794302
  257  4.056717725910233
  263  4.072142508138029
  269  4.08728058437646
  271  4.102362800554971
  277  4.117172774564195
  281  4.131824634971897
  283  4.146424722021268
  293  4.160576342232945