Abstract.
We define the index of composition λ(n) of an integer n ⩾ 2 as λ(n) = log n/log γ(n), where γ(n) stands for the product of the primes dividing n, and first establish that λ and 1/λ both have asymptotic mean value 1. We then establish that, given any ɛ > 0 and any integer k ⩾ 2, there exist infinitely many positive integers n such that . Considering the distribution function F(z,x) := #{n < x : λ(n) > z}, we prove that, given 1 < z < 2 and ɛ > 0, then, if x is sufficiently large,
this last inequality also holding if z ⩾ 2. We then use these inequalities to obtain probabilistic results and we state a conjecture. Finally, using (*), we show that the probability that the abc conjecture does not hold is 0.
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Research supported in part by a grant from NSERC.
Reçu le 17 décembre 2001; en forme révisée le 23 mars 2002 Publié en ligne le 11 octobre 2002
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De Koninck, JM., Doyon, N. À propos de l’indice de composition des nombres. Monatsh. Math. 139, 151–167 (2003). https://doi.org/10.1007/s00605-002-0493-0
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DOI: https://doi.org/10.1007/s00605-002-0493-0