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## The index of composition of an integer

### Jean-Marie De Koninck

Laval University

###
Thursday, April 26, 2007

007 Kemeny Hall, 4 pm

Tea 3:30 pm, 300 Kemeny Hall

**Abstract: ** For each integer *n* at least 2, let
*\gamma(n)* denote the product of the primes which divide
*n*. For example, *\gamma(12)=6*. We define
*\lambda(n)*, ``the index of composition" of *n*, as the
logarithm of *n* in the base *\gamma(n)*, that is,
*\lambda(n):=(\log n)/(\log \gamma(n))*. In a sense,
*\lambda(n)* measures the level of compositeness of
*n*. Although *\lambda(n)* can be arbitrarily large, one can
show that the average value of *\lambda(n)* is 1, that is, the
average over the first *x* integers tends to 1 as *x* tends
to infinity. The study of the local behavior of *n* is more
intricate: for example, can two consecutive values of *\lambda*
be both arbitrarily large? This is related to the famous *abc*
conjecture.

This talk will be accessible to graduate students.