# Computation Workshop Solution Checker

## Recursively Defined Sequence

A sequence $$(t(0), t(1), t(2), t(3), \ldots) = (2,1,5,13, \ldots)$$ begins with $$t(0)=2$$ and $$t(1)=1$$. Then

• $$t(2) = t(1) + 2t(0)$$,
• $$t(3) = t(2) + 2t(1) + 3t(0)$$,
• $$t(4) = t(3) + 2t(2) + 3t(1) + 4t(0)$$,
• $$\ldots$$ and so on.
In general: $$t(n+1) = t(n) + 2t(n-1) + 3t(n-2) + ... + (n+1)t(0)$$ for all positive integers $$n$$.

Part A Determine the last $$9$$ digits of $$t(2021)$$. Determine the last $$9$$ digits of $$t(2021^{2021})$$.

## Divisor Sums

For any positive integer $$n$$, let $$D(n)$$ be the sum of all positive integer divisors of $$n$$ (including $$1$$ and $$n$$ itself). Let $$P(n)$$ be the sum of $$D(i)$$ for all $$i = 1,2,3, \ldots ,n$$. So for example $$P(3) = D(1)+D(2)+D(3) = 1+3+4 = 8$$.

Part A What is $$P(123)$$? What is $$P(1234567)$$?