The following are the first fifty terms of a sequence: 5, 13, 17, 25, 29, 37, 41, 53, 61, 65, 73, 85, 89, 97, 101, 109, 113, 125, 137, 145, 149, 157, 169, 173, 181, 185, 193, 197, 205, 221, 229, 233, 241, 257, 265, 269, 277, 281, 289, 293, 305, 313, 317, 325, 337, 349, 353, 365, 373, 377, . . . Same idea as the other sequences. Describe the pattern of the sequence. If you use a search engine, please don't submit it in the comments. This is a tough one. So I'll start it off with a hint. You might have noticed this. This sequence is a subsequence of last week's sequence. Furthermore, for any i=1, 2, 3, . . ., [a(i)]2 is in the sequence as well. The sequence is a list, not a function, so the index is not related to its corresponding term. Solving the pattern will involve saying why the absent terms from last week's sequence. Additionally, you might recognize the significance of this sequence, apart from figuring why there are fewer terms than last week's sequence. If that's the case, it's fine, just state the significance of the numbers and that counts as solving it. I'll post the solution on next week's sequence. Last week's sequence (0, 1, 2, 4, 5, 8, 9, 10, 13, 16, . . .) was the list of integers who are the sum of two squares. E.g., 10=12+32. E.g., 16=02+42. E.g., 25=32+42. It was not solved. Update: Hint: There are two ways of finding the pattern of the sequence as I said above. So I'll give hints for each direction. The first hint is for if you're trying to solve it by finding out why the members of last week's sequence are excluded: Denote last week's sequence {a(n)} and this week's sequence {b(m)}. For each i=1, 2, 3, . . ., a(i)=c2+d2 for some nonnegative integers c and d. Whether or not a(i) is an element of {b(m)} is determined by how c and d relate to one another. (Don't let the case where c=1 fool you!) From the other direction, if you're trying to find the pattern of the sequence by recognizing the significance of the terms: Euclid's formula generates these numbers. Another hint is to suggest to look at the properties of the square of each term in this sequence and the square of each term in last week's sequence that is not in this list. You might be able to identify that way what the significance is of these numbers.
Monday, June 29, 2009
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2 comments:
I'll take a first stab at it, and if anyone is interested we can discuss it.
The sequence is composed of some primes and of all the products of the terms in the sequence (e.g.: 5, 13, 17, 5*5 = 25, ..., 5*13 = 65). It increases monotonically and in increments that are multiples of 4 (primes that do not fit this pattern is dropped, e.g.: 7 or 11 compared to a(1) = 5).
I am not sure why a(1) = 5. The sequence cannot start with 0, 2 or 4 since then no primes would be included and the sequence would not be very interesting. The all above might give a reason why a(1) is not 3, since the sequence would become
3,7,11,19
and we would miss 21, a product of 3 and 7, because it is only 2 away from 19. And since it has to include all products of terms, 21 must be included, but would violate the increment-by-4 pattern.
Why is a(1) not 1? I don't know.
And according to the hint, if this list is made up of sums of squares, I can only imagine that one of the terms in the sum is a power of 2 (i.e.: 2^2, 4^2...) in order to get the differences in multiples of 4.
Any other thoughts, anyone?
I looked again at the formula that generates these numbers. It then became intuitive for me that the differences between each of the terms is a multiple of 4. That fact follows from the definition.
Good observation!
That said, I don't think it'll help you come up with the solution.
I'm still guessing that the new hint will be enough for you to solve it.
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