Problem to be solved using R.

There is a line of 500000 switches, each initially switched to "on". One by one, 250000 people flip some of the switches.

* Person 1 flips off switch 1.

* Person 2 flips off switches 4, 6, 8, ..., 499998, 500000

* Person 3 flips off switches 6, 9, 12, ..., 499996, 499999 (any switch already off stays off)

* Person 4 flips off switches 8, 12, 16, ...., 499996, 500000 (any switch already off stays off)

* etc.

* In other words, after person 1, person `i` flips off switch numbers `seq(from=2*i,to=500000,by=i)`

Determine which switches remain on after all people have completed their flips.

Hint: work with a vector called `switches`. We'll let element `i` represent the status of the `i`-th switch: 1 for on and 0 for off. Define `switches` to be 500000 1s. Left-arrow the first element to be 0 (to represent the first person turning the first switch to off). Use a `for` loop to "turn off" the switches flipped off by person 2, 3, ..., 250000. Sanity checks: `switches[5979:5989]` gives 0 0 1 0 0 0 0 0 1 0 0 and `sum(switches[32943:39999])` will give 672.

a) What is the total number of switches that remain on?

b) What is the largest switch number (1-500000) that remains on?

c) Print to the screen the first 20 or so numerical values of the switch numbers that are on. This sequence of numbers is a very famous one! What is it?

Walkthrough if you need it:

Step 1: We'll use a vector called `switches` to represent the status of each switch (1 = on, 0 = off). Initialize this vector to be `rep(1,5e5)` ("all on"), then change `switches[1]` to 0 to represent the action of person 1.

Step 2: Write a line of code that "left-arrows" elements 4, 6, 8, ... 499998, 500000 of `switches` to 0 to represent what person 2 does to the switches.

Step 3: Write a line of code that "left-arrows" elements 6, 9, 12, ... 499996, 499999 of `switches` to 0 to represent what person 3 does to the switches.

Step 4: Verify that first 25 elements of `switches` is now `0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1`

Step 5: Person 15843 will turn switches 31686 47529 63372 79215 95058 to "off". Left-arrow `i` to be 15843, then verify that `seq(from=2*i,to=100000,by=i)` indeed produces this sequence of numbers.

Step 6: Synthesize what you have done in Steps 1 to 5 - initialize `switches` to be a vector of 500000 1s (then change the first element to 0), then write a `for` loop to simulate the flips of persons 2, 3, ..., 250000