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Magazine Problem 60
Let \(\tau\) denote the tau function. Prove each. \(\tau(n)\) is odd if and only if \(n\) is a square.
Problem 61
Let \(t\) denote the tau function. Prove each. If \(m\) and \(n\) are relatively prime numbers, then \(\tau(m n)=t(m) \cdot t(n)\)
Problem 61
The number of surjections that can be defined from a finite set \(A\) to a finite set \(B\) is given by \(r ! S(n, r),\) where \(|A|=n\) and \(|B|=r .\) Compute the number of possible surjections from \(A\) to \(B\) if: $$|A|=n,|B|=2$$
Problem 61
Let \(\tau\) denote the tau function. Prove each. If \(m\) and \(n\) are relatively prime numbers, then \(\tau(m n)=\tau(m) \cdot \tau(n)\).
Problem 62
Let \(\tau\) denote the tau function. Prove each. The \(\mathrm{\text{harmonic mean}}\) \(m\) of the numbers \(a_{1}, a_{2}, \ldots, a_{n}\) is the reciprocal of the arithmetic mean of their reciprocals; that is,$$\frac{1}{m}=\frac{1}{n} \sum_{i=1}^{n}\left(\frac{1}{a_{i}}\right)$$ Prove that the harmonic mean of the positive factors of a perfect number \(N\) is an integer. (Hint: If \(d\) is a factor of \(N,\) then so is \(N / d .\) ) (R. Euler, 1987 )
Problem 62
The number of surjections that can be defined from a finite set \(A\) to a finite set \(B\) is given by \(r ! S(n, r),\) where \(|A|=n\) and \(|B|=r .\) Compute the number of possible surjections from \(A\) to \(B\) if: $$|A|=n,|B|=3$$
Problem 62
The harmonic mean \(m\) of the numbers \(a_{1}, a_{2}, \ldots, a_{n}\) is the reciprocal of the arithmetic mean of their reciprocals; that is, $$\frac{1}{m}=\frac{1}{n} \sum_{i=1}^{n}\left(\frac{1}{a_{i}}\right)$$ Prove that the harmonic mean of the positive factors of a perfect number \(N\) is an integer. (Hint: If \(d\) is a factor of \(N\) , then so is \(N / d .\) ) (R. Euler, 1987 )
