Power Series Summation

Here is a problem I came up with: Let $P(x)$ be the power series $a_0x^0+a_1x^1+a_2x^2+a_3x^3+\cdots$ defined as $\displaystyle\sum^{\infty}_{i=1}\displaystyle\sum^{\infty}_{j=1}x^{ij}$. There are certain $n$ such that $a_n$ is odd. Find the number of such $n$ satisfying $n<1000$.

Hint: this is a classic problem in disguise.

EXTENSION: What about the power series $\displaystyle\sum^{\infty}_{i=1}\displaystyle\sum^{\infty}_{j=1}x^{i^2j}$ ?

#Algebra #MathProblem #Math

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Oops, how did that tag get on there...

Daniel Liu - 7 years, 7 months ago

This is purely cosmetic remark, but either at least one of indices should start from 0 or $a_{0} = 0$ . Or am I missing something? But this is actually very interesting problem.

hubert światkiewicz - 7 years, 6 months ago

If either index begins with $0$ , then $a_0 = \infty$ .

Ivan Koswara - 7 years, 6 months ago

This is an interesting way to look at this problem in the context of generating functions. Very enjoyable. Thanks for sharing. :)

Bob Krueger - 7 years, 6 months ago

Basically we count the number of times the exponent $n$ can be expressed as $ij$ for some positive integers $i,j$ . But this also means the number of divisors of $n$ ; for any $i$ divisor of $n$ , there is exactly one $j$ satisfying the condition. So now we count the number of integers $n \in [1,999]$ such that it has an odd number of divisors. Now it's the classic problem :)

The second one is interesting. It basically counts the number of squares that divides $n$ , but is there a particular property (that is relatively easy to check instead of this bruteforcing) that is shared by all $n$ having odd number of square divisors (and none by those having even number of square divisors; or the other way around)?

Ivan Koswara - 7 years, 6 months ago

I don't know either. I was hoping one of you smarter people could figure one out.

Daniel Liu - 7 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$