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Omni-MATH / Fix an integer b≥2. Let f(1)=1, f(2)=2, and for each n≥3, define f(n)=nf(d), where d is the number of base-b digits of n. For which values of b does ∑_(n=1)^(∞)(1)/(f(n)) converge?

Problem

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problem

Fix an integer b≥2b \geq 2. Let f(1)=1f(1) = 1, f(2)=2f(2) = 2, and for each n≥3n \geq 3, define f(n)=nf(d)f(n) = n f(d), where dd is the number of base-bb digits of nn. For which values of bb does ∑n=1∞1f(n)\sum_{n=1}^\infty \frac{1}{f(n)} converge?
Plain-text mathematical notation (without MathML)
Fix an integer b≥2. Let f(1)=1, f(2)=2, and for each n≥3, define f(n)=nf(d), where d is the number of base-b digits of n. For which values of b does ∑_(n=1)^(∞)(1)/(f(n)) converge?
Original LaTeX notation
Fix an integer \(b \geq 2\). Let \(f(1) = 1\), \(f(2) = 2\), and for each \(n \geq 3\), define \(f(n) = n f(d)\), where \(d\) is the number of base-\(b\) digits of \(n\). For which values of \(b\) does \(\sum_{n=1}^\infty \frac{1}{f(n)}\) converge?

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Source and history

task_id
cf89b1f4-dd40-5211-853f-f6886fee40f6
task_key
test--cf89b1f4-dd40-5211-853f-f6886fee40f6
task_revision_id
1
config / split
/ test

Official source

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