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[Tutorial] GCD Convolution

Revision en1, by szaranczuk, 2023-02-04 20:59:01

On today's POI training camp round I have learnt a nice technique that could possibly be useful in some number theory problems. I couldn't find any CF article on it, so I think it's fair enough to share it on my own.

Remark on used notation

In some sums I will use an Iverson notation

Problem: Squarefree Function

Let's define a Squarefree Function b(x) for any positive integer x as x divided by a greatest perfect square, that divides x.

For example: b(1)=1, b(2)=2, b(4)=1, b(6)=6, b(27)=3, b(54)=6, b(800)=2

Given an array a of n105 positive integers, where each ai105 compute sum

1i<jnb(aiaj)

Technique: GCD Convolution

You might probably heard about a sum convolution. For two arrays b, and c, it is defined as an array d such that dk=i,jaiaj[i+j=k] If not, it's basically the same thing as a polynomial multiplication. If B(x)=b0+b1x+b2x2+...+bnxn, and C(x)=c0+c1x+c2x2+...+cmxm, then (BC)(x)=d0+d1x+d2x2+...+dn+mxn+m

Tags number theory, gcd, convolution, tutorial

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