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Betrothed numbers

Type of positive integer pairs

In mathematics, specifically number theory, 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 are: (48, 75), (140, 195), (1050, 1925), (1575, 1648), (2024, 2295), (5775, 6128), which (removing brackets) may be sorted by magnitude .

All known pairs of betrothed numbers have opposite parity. Any pair of the same parity must exceed 1013.

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 = 2^31113813313
1270824975 = 3^25^271942467
1467511664 = 2^419599*8059
1530808335 = 3^357*1619903
1579407344 = 2^431^259*1741
1638031815 = 3^4575211109
1727239544 = 2^3267180833
1512587175 = 35^211*1833439

References

Info: Wikipedia Source

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arithmetic-dynamics divisor-function integer-sequences

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