Betrothed numbers or quasi-amicable numbers are two positive

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s such that the

sum Sum most commonly means the total of two or more numbers added together; see addition. Sum can also refer to: Mathematics * Sum (category theory), the generic concept of summation in mathematics * Sum, the result of summation, the additio ...

of the

proper divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

s 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 In number theory, the aliquot sum ''s''(''n'') of a positive integer ''n'' is the sum of all proper divisors of ''n'', that is, all divisors of ''n'' other than ''n'' itself. That is, :s(n)=\sum\nolimits_d. It can be used to characterize the prime ...

of ''n'': an equivalent condition is that σ(''m'') = σ(''n'') = ''m'' + ''n'' + 1, where σ denotes the

sum-of-divisors function In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (includin ...

. The first few pairs of betrothed numbers 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

s 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 number In mathematics, a quasiperfect number is a natural number ''n'' for which the sum of all its divisors (the divisor function ''σ''(''n'')) is equal to 2''n'' + 1. Equivalently, ''n'' is the sum of its non-trivial divisors (that is, its divisors exc ...

s. 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

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External links

* {{Classes of natural numbers Arithmetic dynamics Divisor function Integer sequences