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Omni-MATH / For an integer n, let f₉(n) denote the number of positive integers d≤9 dividing n. Suppose that m is a positive integer and b₁,b₂,…,b_(m) are real numbers such that …

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problem

For an integer nn, let f9(n)f_{9}(n) denote the number of positive integers d≤9d \leq 9 dividing nn. Suppose that mm is a positive integer and b1,b2,…,bmb_{1}, b_{2}, \ldots, b_{m} are real numbers such that f9(n)=∑j=1mbjf9(n−j)f_{9}(n)=\sum_{j=1}^{m} b_{j} f_{9}(n-j) for all n>mn>m. Find the smallest possible value of mm.
Plain-text mathematical notation (without MathML)
For an integer n, let f₉(n) denote the number of positive integers d≤9 dividing n. Suppose that m is a positive integer and b₁,b₂,…,b_(m) are real numbers such that f₉(n)=∑_(j=1)^(m)b_(j)f₉(n−j) for all n>m. Find the smallest possible value of m.
Original LaTeX notation
For an integer $n$, let $f_{9}(n)$ denote the number of positive integers $d \leq 9$ dividing $n$. Suppose that $m$ is a positive integer and $b_{1}, b_{2}, \ldots, b_{m}$ are real numbers such that $f_{9}(n)=\sum_{j=1}^{m} b_{j} f_{9}(n-j)$ for all $n>m$. Find the smallest possible value of $m$.

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Source and history

task_id
abd82a92-17ef-5d3b-acb6-c37c03555f48
task_key
test--abd82a92-17ef-5d3b-acb6-c37c03555f48
task_revision_id
1
config / split
/ test

Official source

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