Betrothed numbers

Betrothed numbers or quasi-amicable numbers are two positive integers such that the sum of the proper divisors of either number is one more than the value of the other number. In other words, (m, n) are a pair of betrothed numbers if s(m) = n + 1 and s(n) = m + 1, where s(n) is the aliquot sum of n: an equivalent condition is that σ(m) = σ(n) = m + n + 1, where σ denotes the sum-of-divisors function.

The first few pairs of betrothed numbers (sequence A005276 in the OEIS) are: (48, 75), (140, 195), (1050, 1925), (1575, 1648), (2024, 2295), (5775, 6128).

All known pairs of betrothed numbers have opposite parity. Any pair of the same parity must exceed 10¹⁰.

References

• Hagis, Peter, jr; Lord, Graham (1977). "Quasi-amicable numbers". Math. Comput. 31: 608–611. doi:10.1090/s0025-5718-1977-0434939-3. ISSN 0025-5718. Zbl 0355.10010. 
• Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. p. 113. ISBN 1-4020-4215-9. Zbl 1151.11300. 
• Sándor, Jozsef; Crstici, Borislav (2004). Handbook of number theory II. Dordrecht: Kluwer Academic. p. 68. ISBN 1-4020-2546-7. Zbl 1079.11001. 

External links

• Weisstein, Eric W. "Quasiamicable Pair". MathWorld. 

Divisibility-based sets of integers
Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic Arithmetic number
Factorization forms	Prime Composite Semiprime Pronic Sphenic Square-free Powerful Perfect power Achilles Smooth Regular Rough Unusual
Constrained divisor sums	Perfect Almost perfect Quasiperfect Multiply perfect Hemiperfect Hyperperfect Superperfect Unitary perfect Semiperfect Practical Erdős–Nicolas
With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird
Aliquot sequence-related	Untouchable Amicable Sociable Betrothed
Other sets	Deficient Friendly Solitary Sublime Harmonic divisor Frugal Equidigital Extravagant