I have been studying the set of weird numbers, the first 10,000 elements of which are given by https://oeis.org/A006037/b006037.txt, and I noticed something about their distribution that I cannot explain. First, some background:

It has been pointed out on this forum at Are weird numbers more rare than prime numbers? that weird numbers become more dense as you consider more natural numbers. For example the 10th weird number is 10792, the 85th is 42070, and the 10000th is 6527570. The proportion of weird numbers in the first 10792 natural numbers is .0009266123054 while their proportion in the first 42070 natural numbers is .00202044212. This makes sense because you can make more weird numbers by multiplication with primes. In the first 6527570 naturals, however, the proportion of weirds is .0015319637. This is a significant decrease in density from the density within the first 42000 naturals.

I graphed the density of weirds within the naturals for the first 10,000 weirds and saw a relatively steady decrease in density between the 300th and 10000th weird numbers. Around the neighborhood of the 10000th weird number, however, their density begins to increase again. With an algorithm, it was found that there are 86211725 weird numbers in the first 50000000000 natural numbers, giving a density of .0017242345.

My first question is, what is the reason for this increase, then decrease until almost the 10000th weird number, then increase in density towards infinity? In other words, why do weirds become more dense in the beginning of the natural numbers, less dense for a long section of them, and then increasingly more dense? The decrease makes sense to me since many more weird numbers can be made from prime multiples of primitive ones and primes become rarer throughout the set of naturals. The increase in density is what puzzles me.

My second question is, what implication does this density fluctuation have on asymptotic density? In 1974, Erdős proved that the weird numbers have positive natural density. This is a link to his paper: https://old.renyi.hu/~p_erdos/1974-24.pdf Has anyone found a counting function for the number of weird numbers less than or equal to a given x?

Finally, what implication does this have for the distribution of semiperfect numbers? It is known that perfect numbers have asymptotic density 0, therefore I would think that within the set of abundant numbers, weird numbers and semiperfect numbers are complimentary in density. If so, do semiperfect numbers become less dense as one approaches infinity? This is also odd to me because semiperfect numbers can be made from integer multiples of other semiperfect numbers. Is there a limit to the densities of the weird and semiperfect numbers?

Thank you in advance.

  • $\begingroup$ Hi and welcome to MSE. I originally put in the link you already provided, so I've removed that now. Good luck with getting some useful comments and answers here. $\endgroup$ – John Omielan Dec 28 '18 at 2:17
  • $\begingroup$ @JohnOmielan Thank you very much! $\endgroup$ – Gnumbertester Dec 28 '18 at 2:24

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