## Missouri Journal of Mathematical Sciences

### Families of Values of the Excedent Function $\sigma (n) - 2n$

#### Abstract

The excedent function, $e(n) := \sigma(n) - 2n$, measures the amount by which the sum of the divisors of an integer exceeds that integer. Despite having been in the mathematical consciousness for more than $2000$ years, there are many unanswered questions concerning the function. Of particular importance to us is the question of explaining and classifying values in the image of $e(n)$ — especially in understanding the small'' values. We look at extensive calculated data, and use them as inspiration for new results, generalizing theorems in the literature, to better understand a family of values in this image.

#### Article information

Source
Missouri J. Math. Sci., Volume 27, Issue 1 (2015), 37-46.

Dates
First available in Project Euclid: 3 December 2015

https://projecteuclid.org/euclid.mjms/1449161366

Digital Object Identifier
doi:10.35834/mjms/1449161366

Mathematical Reviews number (MathSciNet)
MR3431114

Zentralblatt MATH identifier
06555649

Subjects
Primary: 11A25: Arithmetic functions; related numbers; inversion formulas

#### Citation

Dean, Raven; Erdman, Rick; Klyve, Dominic; Lycette, Emily; Pidde, Melissa; Wheel, Derek. Families of Values of the Excedent Function $\sigma (n) - 2n$. Missouri J. Math. Sci. 27 (2015), no. 1, 37--46. doi:10.35834/mjms/1449161366. https://projecteuclid.org/euclid.mjms/1449161366

#### References

• A. Anavi, P. Pollack, and C. Pomerance, On congruences of the form $\sigma (n) \equiv a \pmod n$, International Journal of Number Theory, 9.1 (2013), 115–124.
• D. M. Burton, Elementary Number Theory, Tata McGraw-Hill Education, 2006.
• P. Cattaneo, Sui numeri quasiperfetti, Boll. Un. Mat. Ital, 3.6 (1951), 59–62.
• Y.-G. Chen and Q.-Q. Zhao, Nonaliquot numbers, Publ. Math. Debrecen, 78.2 (2011), 439–442. MR 2796778 (2011m:11014).
• G. L. Cohen, Generalized quasiperfect numbers, Bulletin of the Australian Mathematical Society, 27.1 (1983), 153–155.
• N. Davis, D. Klyve, and N. Kraght, On the difference between an integer and the sum of its proper divisors, Involve, 6.4 (2013), 493–504. MR 3115982.
• P. Erdős, Über die Zahlen der Form $\sigma(n) - n$ and $n - \phi (n)$, Elem. Math., 28 (1973), 83–86. MR 0337733 (49 #2502).
• L. Euler, Specimen de usu observationum in mathesi pura (E256), Novi Commentarii academiae scientiarum Petropolitanae, 6 (1761), 185–230.
• P. Hagis, Jr. and G. L. Cohen, Some results concerning quasiperfect numbers, J. Austral. Math. Soc. Ser. A, 33.2 (1982), 275–286. MR 668448 (84f:10008).
• P. Pollack and C. Pomerance, On the distribution of some integers related to perfect and amicable numbers, Colloq. Math., 130 (2013), 169–182.
• P. Pollack and V. Shevelev, On perfect and near-perfect numbers, Journal of Number Theory, 132.12 (2012), 3037–3046.
• C. Pomerance, On the congruences $\sigma(n) \equiv a \pmod n$ and $n \equiv a \pmod {\varphi (n)}$, Acta Arith., 26.3 (1974/75), 265–272. MR 0384662 (52 #5535).