Friendly Numbers

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A positive integer \(n\) is called friendly if there exist two positive integers \(j\) and \(k\) such that:

  • \(j+k= n\)

  • There exist two sequences \(\{a_i\}_{i=0}^{\infty} \) and \(\{b_i\}_{i=0}^{\infty}\) of integers (not necessarily positive) satisfying the following conditions. \[\begin{cases} a_i=0 & \text{for all } i>j \\ b_i=0 &\text{for all } i>k \\ \displaystyle \sum_{i=0}^{v} a_i b_{v-i} = 0 & \text{for all } v \in \{1, 2, \cdots , n-1 \} \\ a_0b_0= a_jb_k= 1 \\ \end{cases}\]

Find the number of positive integers \(\leq 1000\) which are not friendly.

Details and assumptions

  • As an explicit example, when \(n=3\) and \(j=1, k=2\), the last condition reads: \[\begin{cases} a_i=0 & \text{for all } i>1 \\ b_i=0 & \text{for all } i>2 \\ a_0b_0= a_1b_2= 1 \\ a_0b_1 + a_1b_0= 0 \\ a_0b_2 + a_1b_1 + a_2b_0= 0 \end{cases} \] It isn't hard to verify that the sequences \(\{a_0, a_1, a_2, a_3, \cdots \} = \{1, 1, 0, 0, \cdots \} \) and \(\{b_0, b_1, b_2, b_3, \cdots \} = \{1, -1, 1, 0, 0, \cdots \} \) satisfy all the conditions above, so \(3\) is a friendly number.

  • By definition, \(1\) and \(2\) are unfriendly numbers.

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