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By taking a look at the first few weird numbers: $$(70, 836, 4030, 5830, 7192, 7912, 9272, 10430)$$ It is certain that prime numbers occurs more often within this range of numbers.

But are weird numbers more rare than prime numbers in the long run? Sure, by the definition of infinity, there are infinite prime numbers and infinite weird numbers. But if you calculated prime numbers and weird numbers for a finite amount of time, would prime numbers be more common than weird numbers?

This may not be very easy to explain, but I'd appreciate an attempt to keep it as simple as possible.

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  • $\begingroup$ Almost surely primes have a greater density but a proof is lacking. I don't know of any growth result for the weird numbers. $\endgroup$ – Balarka Sen Jul 28 '14 at 10:14
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    $\begingroup$ I had no idea such a thing as weird numbers existed. According to your explanation of comparison of the sizes of those sets of numbers, you seem to be looking for the asymptotic distribution of weird numbers. $\endgroup$ – Git Gud Jul 28 '14 at 10:14
  • $\begingroup$ In any case, why would one be interested in comparison of growth rates of primes and weird numbers at all? There is no apparent relation between them that I could think of. Why should it be of general interest? $\endgroup$ – Balarka Sen Jul 28 '14 at 10:18
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    $\begingroup$ Hmm. the link you added a few minutes ago contains a claim that "the sequence of weird number has positive Schnirelmann density". This basically answer your own question. $\endgroup$ – achille hui Jul 28 '14 at 11:47
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    $\begingroup$ @BalarkaSen, I think people refer to it as "curiosity" $\endgroup$ – Paul Draper Jul 28 '14 at 18:35
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Wikipedia cites Benkoski, Stan; Erdős, Paul (April 1974). "On Weird and Pseudoperfect Numbers" for the fact the weird numbers have positive asymptotic density. But primes have zero asymptotic density, so in a sense, in a long run weird numbers are not only more abundant, but infinitely more abundant. More quantitatively, if we let $w(n)$ be the weird-number-counting function, we should have $w(n)\sim \alpha n$ for some parameter $0<\alpha<1$, whereas the prime number theorem tells us $\pi(n)\sim\frac{n}{\log n}$.

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    $\begingroup$ Unexpected fact I guess. (+1) $\endgroup$ – Balarka Sen Jul 28 '14 at 10:22
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    $\begingroup$ @BalarkaSen It is not that unexpected after all, since given one weird number it is easy to make "many" other weird numbers by multiplication. In fact if one reads the mentioned paper they say that the set of weird numbers which are multiples of $70$ also has positive asymptotic density (and the same with other weird multipliers). $\endgroup$ – Jeppe Stig Nielsen Jul 28 '14 at 10:53
  • $\begingroup$ It'd be interesting to see a bound/estimate for the onset of $w(n)>\pi(n)$. $\endgroup$ – Semiclassical Jul 28 '14 at 11:52
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    $\begingroup$ @Charles: Can you sketch your argument/heuristics for that? $\endgroup$ – Semiclassical Jul 28 '14 at 15:28
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    $\begingroup$ @Semiclassical: Nothing fancy, just estimates/bounds for the reciprocal density of weird numbers and then take exp to get the crossover for the primes. $\endgroup$ – Charles Jul 28 '14 at 16:53
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Wanted to just leave this as a comment but this is probably easier. As you probably know the OEIS usually has an abudance of information for things like this , https://oeis.org/A006037 , just by inspection one can see that the wierd numbers be come more dense as they grow in size. Far from a proof but useful in getting an idea of their denisty.

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