Normal largest gap between prime factors
Journal de Théorie des Nombres de Bordeaux, Tome 31 (2019) no. 3, pp. 747-749.

Désignons par {p j (n)} j=1 ω(n) la suite croissante des facteurs premiers distincts d’un entier n. Nous explicitions les détails de la preuve d’un énoncé d’Erdős impliquant que, pour toute fonction ξ(n) tendant vers l’infini avec n, nous avons

f(n):=max1j<ω(n)loglogpj+1(n)logpj(n)=log3n+O(ξ(n))

pour presque tout entier n.

Let {p j (n)} j=1 ω(n) denote the increasing sequence of distinct prime factors of an integer n. We provide details for the proof of a statement of Erdős implying that, for any function ξ(n) tending to infinity with n, we have

f(n):=max1j<ω(n)loglogpj+1(n)logpj(n)=log3n+O(ξ(n))

for almost all integers n.

Reçu le :
Accepté le :
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DOI : https://doi.org/10.5802/jtnb.1107
Classification : 11N56,  14G42
Mots clés : Distribution of prime factors, normal order, largest gap.
@article{JTNB_2019__31_3_747_0,
     author = {G\'erald Tenenbaum},
     title = {Normal largest gap between prime factors},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {747--749},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {3},
     year = {2019},
     doi = {10.5802/jtnb.1107},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1107/}
}
Gérald Tenenbaum. Normal largest gap between prime factors. Journal de Théorie des Nombres de Bordeaux, Tome 31 (2019) no. 3, pp. 747-749. doi : 10.5802/jtnb.1107. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1107/

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