2019 CIME I Problems/Problem 15
Let be a sequence of functions going from
to
defined recursively by
and
for all
. Compute the greatest integer less than or equal to
.
Solution
Note that the function on the right is ’s sum function, so if
is multiplicative so is
. Now since
is multiplicative, it is not hard to see using induction that all
are multiplicative too.
Therefore we can just consider one prime and its exponent. Say . Note that
and
. Then
by the Hockey Stick Identity. We can continue this process (summing and using the hockey stick identity for each exponent) to obtain
.
Now and
thus our answer is
.
See also
2019 CIME I (Problems • Answer Key • Resources) | ||
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