Originally posted by: DrPizza
wha? Huh??
2,4,6,8,.... the next two answers are 10 and 12
What do you mean "The next two answers ARE..."? I thought you were looking for geniuses or super-geniuses. Geniuses and super-geniuses don't want such an answer dictated to them, especially since that's not the only logical answer. You should have said "one possible solution for the next two terms of the sequence is 10, 12." But, being that it's Easter, I forgive you for that transgression.
Now, allow me to put YOU "in the proper frame of mind.... "
2, 4, 6, 8, 27, 30, 33, 96,... This is the sequence of numbers that are evenly divisible by the number of prime numbers less than that number. i.e. the prime numbers less than 33 are 2,3,5,7,11,13,17,19,23,29,31; that's 11 prime numbers. 33 is divisible by 11. There are 11 prime numbers less than 34, 35, and 36, but none of these are divisible by 11, thus they're not on the list. 37 is a prime, so it of course isn't divisible by any of the primes less than itself, 38 has 12 primes less than it; 38 isn't evenly divisible by 12, etc.
Heck, even when you think you're certain of what comes next...
2, 4, 6, 8, 10, 12, 14, 16, 18,. . . doesn't mean that the next digit HAS to be 20. Thinking outside the box, this beginning of a sequence can be generated such that a(n) is equal to n + the product of non-zero digits in n. In other words, it's generated this way: 1 + 1, 2 + 2, 3 + 3, 4 + 4, 5 + 5, 6 + 6, 7 + 7, 8 + 8, 9 + 9, 10 + 1, 11 + 1*1, 12 + 1*2, ... The hundredth digit in this sequence would be 100 + 1, the 248th digit would be 248 + 2*4*8 = 312.
Since the first sequence I posted had a 27, 30, and 33 in it, how about this one?
2, 4, 6, 8, 10, 12, 14, 16, 27, 30, 33,... This is a simpler sequence. a(n) is the number closest to the nth prime that's also divisible by n. i.e. the 9th prime number is 23. The closest number to 23 that's divisible by 9 is 27, so the 9th term in this sequence is 27.
Now, I've figured out 2 separate solutions to the sequence you posted at the top. However, I figure it'll take a super-genius to realize that my solution is correct. Since your OP contains "To put
your in the proper frame of mind.... ", I shall not assume you are qualified. Thus, while I'll forgive you for the presumption that 10,12 were next in that trivial sequence, I rarely forgive errors involving "you, your, you're, there, their, and they're." I'd PM Marilyn Vos Savant, but I figure that we super-geniuses don't want to be bothered by such simple trivia. Besides, checking the solution is incredibly trivial compared to coming up with the solution.