Document Type : Original Manuscript


1 Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad, Iran.

2 Department of Computer Science, School of Mathematics, Statistics and Computer Science, University of Tehran, P.O. Box 141556619, Tehran, Iran.


‎For a positive integer $m$‎, ‎a subset of divisors of $m$ is called a \textit{divisor topology on $m$} if it contains $1 $ and $m$ and it is closed under taking $\gcd$ and $\rm lcm$‎. ‎If $m=p_1\dots p_n$ is a square free positive integer‎, ‎then a divisor topology $m$ corresponds to a topology on the set $[n]=\{1,2,\ldots,n\}$‎. ‎Giving some facts about divisor topologies‎, ‎we give a recursive formula for the number of divisor topologies on a positive integer‎.


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