For example, if we had an arbitrary event that could either yield the result a or b, we might naively assign a probability of .5 to each result. After several trials, we can use the results of the trials to adjust our expectations of the probabilities of the event yielding a or b. If the first n...
Very cool. So, the longest chain through the first 10,000 integers was 17 and then through 10,000,000 that could only be extended to 21 (I count 21, not 22). Somewhat surprising and interesting that the longest chain is lengthened so slowly, so to speak.
It is well known that the Harmonic Series diverges (1/1+1/2+1/3+1/4+...), but that the series [1/1+1/4+1/9+1/16+...] converges. In the first series, the denominators are the integers, whereas in the second example, the denominators are the integers to the power of 2. My question is, at what...
Here is the process:
For example, take 28. Its prime factors (taken with multiplicity) are 2, 2, and 7. The sum of these is 11. Double it and you get 22. So, the sum of prime factors of 28 multiplied by 2 is 22. Repeat this process until you reach a loop or a stable number.
For 28, if you...