P920
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Tau Numbers

ℹ️Published on Sunday, 8th December 2024, 10:00 am; Solved by 247;
Difficulty rating: 30%

For a positive integer n we define τ(n) to be the count of the divisors of n. For example, the divisors of 12 are {1,2,3,4,6,12} and so τ(12)=6.

A positive integer n is a tau number if it is divisible by τ(n). For example τ(12)=6 and 6 divides 12 so 12 is a tau number.

Let m(k) be the smallest tau number x such that τ(x)=k. For example, m(8)=24, m(12)=60 and m(16)=384.

Further define M(n) to be the sum of all m(k) whose values do not exceed 10n. You are given M(3)=3189.

Find M(16).



Soluzione

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