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. ## Quasi-sociable numbers Quasi-sociable numbers or reduced sociable numbers are numbers whose aliquot sums minus one form a cyclic sequence that begins and ends with the same number. They are generalizations of the concepts of betrothed numbers and quasiperfect numbers. The first quasi-sociable sequences, or quasi-sociable chains, were discovered by Mitchell Dickerman in 1997: * 1215571544 = 2^3*11*13813313 * 1270824975 = 3^2*5^2*7*19*42467 * 1467511664 = 2^4*19*599*8059 * 1530808335 = 3^3*5*7*1619903 * 1579407344 = 2^4*31^2*59*1741 * 1638031815 = 3^4*5*7*521*1109 * 1727239544 = 2^3*2671*80833 * 1512587175 = 3*5^2*11*1833439 ## References * Hagis, Peter, jr; Lord, Graham (1977). "Quasi-Amicable Numbers". Math. Comput. 31 (138): 608–611. doi:10.1090/s0025-5718-1977-0434939-3. ISSN 0025-5718. * Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds (2006). Handbook of Number Theory I. Dordrecht: Springer-Verlag. p. 113. ISBN 978-1-4020-4215-7. * Sándor, Jozsef; Crstici, Borislav (2004). Handbook of Number Theory II. Dordrecht: Kluwer Academic. p. 68. ISBN 978-1-4020-2546-4. https://archive.org/details/handbooknumberth00sand_075. ## External links * Weisstein, Eric W.. "Quasiamicable Pair". http://mathworld.wolfram.com/QuasiamicablePair.html. * v * t * e 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 Base-dependent| * Equidigital * Extravagant * Frugal * Harshad * Polydivisible * Smith Other sets| * Deficient * Friendly * Solitary * 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* Sublime * Superabundant * Superior highly composite * Superperfect * Untouchable | By properties of Ω(n)| * Almost prime * Semiprime By properties of φ(n)| * Highly cototient * Highly totient * Noncototient * Nontotient * Perfect totient * Sparsely totient By properties of s(n)| * Almost perfect * Amicable * Betrothed * Deficient * Perfect * Semiperfect * Sociable Other| * Euclid * Fortunate Dividing a quotient| * Wieferich * Wall–Sun–Sun * Wolstenholme prime * Wilson Other prime factor or divisor related numbers| * Blum * Erdős–Nicolas * Erdős–Woods * Friendly * Giuga * Harmonic divisor * Lucas–Carmichael * Pronic * Regular * Rough * Smooth * Sociable * Sphenic * Størmer * Super-Poulet * Zeisel Binary numbers| * Evil * Odious * Pernicious Generated via a sieve| * Lucky * Prime Sorting related| * Pancake number * Sorting number Natural language related| * Aronson's sequence * Ban Graphemics related| * Strobogrammatic * Mathematics portal 0.00 (0 votes) Original source: 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