## Multiamicable numbers

HTML articles powered by AMS MathViewer

- by Graeme L. Cohen, Stephen F. Gretton and Peter Hagis PDF
- Math. Comp.
**64**(1995), 1743-1753 Request permission

## Abstract:

Multiamicable numbers are a natural generalization of amicable numbers: two numbers form a multiamicable pair if the sum of the proper divisors of each is a multiple of the other. Many other generalizations have been considered in the past. This paper reviews those earlier generalizations and gives examples and properties of multiamicable pairs. It includes a proof that the set of all multiamicable numbers has density 0.## References

- S. Battiato and W. Borho,
*Are there odd amicable numbers not divisible by three?*, Math. Comp.**50**(1988), no. 182, 633–637. MR**929559**, DOI 10.1090/S0025-5718-1988-0929559-5 - R. D. Carmichael,
*Recent Publications: Reviews: History of the Theory of Numbers. Volume I: Divisibility and Primality*, Amer. Math. Monthly**26**(1919), no. 9, 396–403. MR**1519382**, DOI 10.2307/2971917 - Graeme L. Cohen,
*On an integer’s infinitary divisors*, Math. Comp.**54**(1990), no. 189, 395–411. MR**993927**, DOI 10.1090/S0025-5718-1990-0993927-5
—, - Henri Cohen,
*On amicable and sociable numbers*, Math. Comp.**24**(1970), 423–429. MR**271004**, DOI 10.1090/S0025-5718-1970-0271004-6 - L. E. Dickson,
*Amicable Number Triples*, Amer. Math. Monthly**20**(1913), no. 3, 84–92. MR**1517797**, DOI 10.2307/2973442
Leonard Eugene Dickson, - P. Erdös,
*On amicable numbers*, Publ. Math. Debrecen**4**(1955), 108–111. MR**69198** - Achim Flammenkamp,
*New sociable numbers*, Math. Comp.**56**(1991), no. 194, 871–873. MR**1052094**, DOI 10.1090/S0025-5718-1991-1052094-3 - Richard K. Guy,
*Unsolved problems in number theory*, Problem Books in Mathematics, Springer-Verlag, New York-Berlin, 1981. MR**656313** - Peter Hagis Jr.,
*Unitary amicable numbers*, Math. Comp.**25**(1971), 915–918. MR**299551**, DOI 10.1090/S0025-5718-1971-0299551-2 - C. Krishnamurthy,
*Some sets of amicable numbers of higher order*, Indian J. Pure Appl. Math.**11**(1980), no. 12, 1549–1553. MR**617829** - A. Mąkowski,
*On some equations involving functions $\phi (n)$ and $\sigma (n)$*, Amer. Math. Monthly**67**(1960), 668–670. MR**130209**, DOI 10.2307/2310107 - Thomas E. Mason,
*On Amicable Numbers and Their Generalizations*, Amer. Math. Monthly**28**(1921), no. 5, 195–200. MR**1519771**, DOI 10.2307/2973750 - Wayne McDaniel,
*On odd multiply perfect numbers*, Boll. Un. Mat. Ital. (4)**3**(1970), 185–190. MR**0262154** - David Moews and Paul C. Moews,
*A search for aliquot cycles below $10^{10}$*, Math. Comp.**57**(1991), no. 196, 849–855. MR**1094955**, DOI 10.1090/S0025-5718-1991-1094955-5
Paul Poulet, - Herman J. J. te Riele,
*On generating new amicable pairs from given amicable pairs*, Math. Comp.**42**(1984), no. 165, 219–223. MR**725997**, DOI 10.1090/S0025-5718-1984-0725997-0
W. Sierpiñski, - Benjamin Franklin Yanney,
*Another Definition of Amicable Numbers and Some of their Relations to Dickson’s Amicables*, Amer. Math. Monthly**30**(1923), no. 6, 311–315. MR**1520272**, DOI 10.2307/2300268

*Generalisations of amicable numbers*, Internal Report No. 36, School of Mathematical Sciences, University of Technology, Sydney, April 1992.

*History of the theory of numbers*, Vol. 1, Chelsea, New York, 1971.

*La chasse aux nombres*, Bruxelles, 1929.

*Elementary theory of numbers*, North-Holland, Amsterdam, 1988.

## Additional Information

- © Copyright 1995 American Mathematical Society
- Journal: Math. Comp.
**64**(1995), 1743-1753 - MSC: Primary 11A25
- DOI: https://doi.org/10.1090/S0025-5718-1995-1308449-6
- MathSciNet review: 1308449