Abstract.
For a positive integer n we let τ(n) denote the number of its positive divisors. In this paper, we obtain lower and upper bounds for the average value of the ratio τ(n + 1)/τ(n) as n ranges through positive integers in the interval [1,x]. We also study the cardinality of the sets {τ(p − 1) : p ≤ x prime} and {τ(2n − 1) : n ≤ x}.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
WR Alford A Granville C Pomerance (1994) ArticleTitleThere are infinitely many Carmichael numbers Ann Math 139 IssueID2 703–722 Occurrence Handle1283874 Occurrence Handle0816.11005 Occurrence Handle10.2307/2118576
Becheanu M, Luca F, Shparlinski IE (2006) On the sums of complementary divisors. Intern J Number Theory, to appear
NG de Bruijn (1966) ArticleTitleOn the number of positive integers ≤ x and free prime factors > y. II Indag Math 28 239–247 Occurrence Handle205945
J-M Deshouillers H Iwaniec (1982) ArticleTitleAn additive divisor problem J London Math Soc 26 1–14 Occurrence Handle10.1112/jlms/s2-26.1.1 Occurrence Handle667238 Occurrence Handle0462.10030
P Erdös L Mirsky (1952) ArticleTitleThe distribution of values of the divisor function d(n) Proc London Math Soc 2 257–271 Occurrence Handle10.1112/plms/s3-2.1.257 Occurrence Handle49932 Occurrence Handle0047.04602
RR Hall G Tenenebaum (1988) Divisors Univ Press Cambridge Occurrence Handle0653.10001
GH Hardy S Ramanujan (1917) ArticleTitleThe normal number of prime factors of an integer Quart J Math (Oxford) 48 76–92 Occurrence HandleJFM 46.0262.03
G Harman (2005) ArticleTitleOn the number of Carmichael numbers up to x Bull London Math Soc 37 641–650 Occurrence Handle10.1112/S0024609305004686 Occurrence Handle2164825 Occurrence Handle1108.11065
Khripunova MB (1998) Additive problems for integers with a given number of prime divisors. Math Notes 63: 658–669; translation from Mat Zametki 63: 749–762 (in Russian)
W Ljunggren (1943) ArticleTitleSome theorems on indeterminate equations of the form \(\frac{x^n-1}{x-1}=y^q\) Norsk Math Tiddskr 25 17–20 Occurrence Handle18674 Occurrence Handle0028.00901
Luca F, Pomerance C (2007) On the range of some classical arithmetic functions. Preprint
G Tenenbaum (1995) Introduction to Analytic and Probabilistic Number Theory Univ Press Cambridge
Timofeev NM, Khripunova MB (2000) On the difference between the number of prime divisors from subsets for consecutive numbers. Mat Zametki 68: 725–738; translation in Math Notes 68: 614–626
Author information
Authors and Affiliations
Additional information
Authors’ addresses: Florian Luca, Instituto de Matemáticas, Universidad Nacional Autónoma'de'México, C.P. 58089, Morelia, Michoacán, México; Igor E. Shparlinski, Department of Computing, Macquarie University, Sydney, NSW 2109, Australia
Rights and permissions
About this article
Cite this article
Luca, F., Shparlinski, I. On the values of the divisor function. Monatsh Math 154, 59–69 (2008). https://doi.org/10.1007/s00605-007-0511-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-007-0511-3