| Title:
|
Bounds for the counting function of the Jordan-Pólya numbers (English) |
| Author:
|
De Koninck, Jean-Marie |
| Author:
|
Doyon, Nicolas |
| Author:
|
Razafindrasoanaivolala, A. Arthur Bonkli |
| Author:
|
Verreault, William |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
56 |
| Issue:
|
3 |
| Year:
|
2020 |
| Pages:
|
141-152 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A positive integer $n$ is said to be a Jordan-Pólya number if it can be written as a product of factorials. We obtain non-trivial lower and upper bounds for the number of Jordan-Pólya numbers not exceeding a given number $x$. (English) |
| Keyword:
|
Jordan-Pólya numbers |
| Keyword:
|
factorial function |
| Keyword:
|
friable numbers |
| MSC:
|
11A41 |
| MSC:
|
11A51 |
| MSC:
|
11B65 |
| MSC:
|
11N05 |
| idZBL:
|
Zbl 07250675 |
| idMR:
|
MR4156441 |
| DOI:
|
10.5817/AM2020-3-141 |
| . |
| Date available:
|
2020-09-02T08:49:36Z |
| Last updated:
|
2020-11-23 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148292 |
| . |
| Reference:
|
[1] De Angelis, V.: Stirling’s series revisited.Amer. Math. Monthly 116 (2009), 839–843. MR 2572092, 10.4169/000298909X474918 |
| Reference:
|
[2] De Koninck, J.-M., Luca, F.: Analytic Number Theory: Exploring the Anatomy of Integers.Graduate Studies in Mathematics, vol. 134, American Mathematical Society, Providence, Rhode Island, 2012. MR 2919246 |
| Reference:
|
[3] Ellison, W., Ellison, F.: Prime Numbers.Hermann, Paris, 1985. MR 0814687 |
| Reference:
|
[4] Ennola, V.: On numbers with small prime divisors.Ann. Acad. Sci. Fenn. Ser. AI 440 (1969), 16 pp. MR 0244175 |
| Reference:
|
[5] Erdös, P., Graham, R.L.: On products of factorials.Bull. Inst. Math. Acad. Sinica 4 (2) (1976), 337–355. MR 0460262 |
| Reference:
|
[6] Feller, W.: An introduction to probability theory and its applications.Vol. I, Third edition, John Wiley $\&$ Sons, Inc., New York-London-Sydney, 1968, xviii+509 pp. MR 0228020 |
| Reference:
|
[7] Granville, A.: Smooth numbers: computational number theory and beyond. Algorithmic number theory: lattices, number fields, curves and cryptography.Math. Sci. Res. Inst. Publ. 44 (2008), 267–323, Cambridge Univ. Press, Cambridge. MR 2467549 |
| Reference:
|
[8] Luca, F.: On factorials which are products of factorials.Math. Proc. Cambridge Philos. Soc. 143 (3) (2007), 533–542. MR 2373957, 10.1017/S0305004107000308 |
| Reference:
|
[9] Nair, S.G., Shorey, T.N.: Lower bounds for the greatest prime factor of product of consecutive positive integers.J. Number Theory 159 (2016), 307–328. MR 3412724, 10.1016/j.jnt.2015.07.014 |
| Reference:
|
[10] Rosser, J.B.: The $n$-th prime is greater than $n\log n$.Proc. London Math. Soc. (2) 45 (1938), 21–44. MR 1576808 |
| Reference:
|
[11] Rosser, J.B., Schoenfeld, L.: Approximate formulas for some functions of prime numbers.Illinois J. Math. 6 (1962), 64–94. MR 0137689, 10.1215/ijm/1255631807 |
| Reference:
|
[12] Tenenbaum, G.: Introduction à la théorie analytique des nombres.Collection Échelles, Belin, 2008. MR 0675777 |
| . |
