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 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 prim ...

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). 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 excl ...

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