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 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

Classes of natural numbers

Powers and
related numbers

Of the form
a × 2^b ± 1

Other polynomial
numbers

Recursively defined
numbers

Possessing a
specific set
of other numbers

Expressible via
specific sums

Generated
via a sieve

Lucky

Code related

Meertens

Figurate numbers

2-dimensional

centered	Centered triangular Centered square Centered pentagonal Centered hexagonal Centered heptagonal Centered octagonal Centered nonagonal Centered decagonal Star

non-centered	Triangular Square Square triangular Pentagonal Hexagonal Heptagonal Octagonal Nonagonal Decagonal Dodecagonal

3-dimensional

centered	Centered tetrahedral Centered cube Centered octahedral Centered dodecahedral Centered icosahedral

non-centered	Tetrahedral Octahedral Dodecahedral Icosahedral Stella octangula

pyramidal	Square pyramidal Pentagonal pyramidal Hexagonal pyramidal Heptagonal pyramidal

4-dimensional

centered	Centered pentachoric Squared triangular

non-centered	Pentatope

Pseudoprimes

Combinatorial
numbers

Arithmetic functions

By properties of σ(n)	Abundant Almost perfect Arithmetic Colossally abundant Descartes Hemiperfect Highly abundant Highly composite Hyperperfect Multiply perfect Perfect Practical number Primitive abundant Quasiperfect Refactorable Sublime Superabundant Superior highly composite Superperfect

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)	Amicable Betrothed Deficient Semiperfect

Dividing a quotient

Other prime factor
or divisor related
numbers

Recreational
mathematics

Base-dependent
numbers

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