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I want to find a function such that $$ \sum_{0<j<n/k } f(kj)=1 $$ Where the sum j is taken over the natural numbers, And the series is satisfied for all integers k and n, I was thinking of trying to turn it into a telescoping series, but that didnt work out, so maybe there is some oscillatory function that behaves like this? Can someone help me? If someone cant find a function that behaves exactly like this, can someone find a function that behaves like this for large n.

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Do you want a single function $f$ for which your sum is 1 for any $n, k$? That's impossible. Do you perhaps want $f$ to be a collection of functions, one for each $n$? –  Rick Decker Nov 6 '12 at 1:37
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In particular, for $k=1$, your sum is $f(1)+f(2)+\cdots+f(n-1)$. There's only one way to make that 1 for all $n$, and that's to take $f(1)=1$, $f(n)=0$ for $n\gt1$. And that won't work for $k\gt1$. –  Gerry Myerson Nov 6 '12 at 2:12

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