Gusty Games

 

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 \(t(2021)\).
Part B 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 \(P(123)\)?
Part B What is \(P(1234567)\)?