Abstract. For natural numbers n we inspect all factorizations n = ab of n with \documentclass[12pt]{minimal}
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$a \le b$\end{document} in \documentclass[12pt]{minimal}
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$\Bbb N$\end{document} and denote by \documentclass[12pt]{minimal}
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$n=a_n b_n$\end{document} the most quadratic one, i.e. such that \documentclass[12pt]{minimal}
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$b_n - a_n$\end{document} is minimal. Then the quotient \documentclass[12pt]{minimal}
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$\kappa (n) := a_n/b_n$\end{document} is a measure for the quadraticity of n. The best general estimate for \documentclass[12pt]{minimal}
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$\kappa (n)$\end{document} is of course very poor: \documentclass[12pt]{minimal}
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$1/n \le \kappa (n)\le 1$\end{document}. But a Theorem of Hall and Tenenbaum [1, p. 29], implies$(\log n)^{-\delta -\varepsilon } \le \kappa (n) \le (\log n)^{-\delta }$ on average, with \documentclass[12pt]{minimal}
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$\delta = 1 - (1+\log _2 \,2)/\log 2=0,08607 \ldots $\end{document} and for every \documentclass[12pt]{minimal}
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$\varepsilon >0$\end{document}. Hence the natural numbers are fairly quadratic.¶\documentclass[12pt]{minimal}
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$\kappa (n)$\end{document} characterizes a specific optimal factorization of n. A quadraticity measure, which is more global with respect to the prime factorization of n, is \documentclass[12pt]{minimal}
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$\kappa ^*(n):= \textstyle\sum\limits \limits _{1\le a \le b, ab=n} a/b$\end{document}. We show $\kappa ^*(n) \sim \frac {1}{2}$ on average, and $\kappa ^*(n)=\Omega (2^{\frac {1}{2}(1-\varepsilon ) {\log}\, n/{\log} _2n})$for every \documentclass[12pt]{minimal}
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$\varepsilon>0$\end{document}.
