Is it true that for all positive integers n, the inequality $\displaystyle d(n)\leq 1 + log_2 n$ holds?
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Originally Posted by alexmahone Is it true that for all positive integers n, the inequality $\displaystyle d(n)\leq 1 + log_2 n$ holds? I assume from the title that $\displaystyle d(n)$ is the number of divisors of $\displaystyle n$. Does that hold for $\displaystyle n=144~?$
Originally Posted by Plato I assume from the title that $\displaystyle d(n)$ is the number of divisors of $\displaystyle n$. Does that hold for $\displaystyle n=144~?$ 144 = 12^2 d(144) = 2 + 1 = 3 $\displaystyle 1 + log_2 144 \approx 1 + 7.17 = 8.17$ It does hold for n = 144.
Uh... no! 1 2 3 4 6 8 9 12 16 ... all divide 144, so the left hand side is larger than the right hand side
Originally Posted by TheChaz Uh... no! 1 2 3 4 6 8 9 12 16 ... all divide 144, so the left hand side is larger than the right hand side Oops ... I should have done: 144 = 2^4 * 3^2 d(n) = (4 + 1)(2 + 1) = 15
Originally Posted by alexmahone 144 = 12^2 d(144) = 2 + 1 = 3 $\displaystyle 1 + log_2 144 \approx 1 + 7.17 = 8.17$ It does hold for n = 144. Because $\displaystyle 144=2^4\cdot 3^2$ then $\displaystyle d(144)=(4+1)(3+1)=15$.
Perhaps there is a mistake in the book: Is $\displaystyle d(n)\geq 1 + log_2 n$ true?
Originally Posted by alexmahone Perhaps there is a mistake in the book: Is $\displaystyle d(n)\geq 1 + log_2 n$ true? The divisor bound « What’s new
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