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How

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Let $$a_1,a_2,\ldots$$ be a sequence determined by the rule $$a_n= \frac{a_{n-1}}{2}$$ if $$a_{n-1}$$ is even and $$a_n=3a_{n-1}+1$$ if $$a_{n-1}$$ is odd. For how many positive integers $$a_1 \le 2008$$ is it true that $$a_1$$ is less than each of $$a_2,a_3,and$$ $$a_4$$?

Jun 19, 2019

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I think there are 502 that satisfy the condition.

Only the odd numbers provide a2 greater than a1.  There are 1004 of them.

All these a2's are even, so a3 = a2/2 = (3a1 + 1)/2 which is greater than a1.

The corresponding a4's must alternate even and odd.  The even ones will be a3/2 = (3a1 + 1)/4 which must be less than or equal to a1 (equality when a1 = 1).  The odd ones must be   3(3a1 + 1)/2 + 1 = 9a1/2 + 5/2 > a1.  There are 1004/2 = 502 of these.

Jun 19, 2019