Question 27

For a positive integer $$n$$, let $$d(n)$$ denote the number of positive divisors of $$n$$. For example, $$d(4) = 3$$, The smallest positive integer $$n$$ for which $$d(n-2)+d(n)+d(n+2)=21$$ is

Correct Answer: 38

Solution

Checking the positive integers below $$38$$ gives no total of $$21$$ for the three divisor counts. At $$n=38$$, the numbers are $$36,38,40$$. Their divisor counts are $$d(36)=9$$, $$d(38)=4$$ and $$d(40)=8$$, whose sum is $$21$$.

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For a positive integer $$n$$, let $$d(n)$$ denote the number of positive divisors of $$n$$. For example, $$d(4) = 3$$, The smallest positive integer $$n$$ for which $$d(n-2)+d(n)+d(n+2)=21$$ is

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