Engaged numbers or quasi-friendly numbers are two positive integers , for which the sum of the proper divisors of each number is 1 more than the second number. In other words, ( m , n ) is a pair of betrothed numbers if s ( m ) = n + 1 and s ( n ) = m + 1, where s ( n ) is the sum of the proper dividers of the number n ( aliquot sum of n ). Equivalent condition is Ο 1 ( m ) = Ο 1 ( n ) = m + n + 1, where Ο 1 ( n ) is the sum of all divisors of n .
The first few pairs of engaged numbers that make up the sequence A005276 in OEIS are (48, 75), (140, 195), (1050, 1925), (1575, 1648), (2024, 2295), (5775, 6128).
Not important for the theory of numbers , but are an interesting element of entertaining mathematics .
Content
Facts
- All known pairs of betrothed numbers have opposite parity . It is not known whether there is a pair of betrothed numbers of the same parity. Any pair of the same parity must exceed 10 10 .
- Sometimes slightly redundant numbers are considered a particular case of betrothed numbers, like numbers engaged to themselves.
- It is not known whether or not the number of pairs of betrothed numbers is infinite.
See also
- Friendly numbers
- Slightly redundant numbers
- Redundant numbers
- Social numbers
Sources
- Hagis, Peter, jr; Lord, Graham. Quasi-amicable numbers (English) // Math. Comput. : journal. - 1977. - Vol. 31 . - P. 608-611 . - ISSN 0025-5718 . - DOI : 10.1090 / s0025-5718-1977-0434939-3 .
- Handbook of number theory I. - Dordrecht: Springer-Verlag , 2006. - P. 113. - ISBN 1-4020-4215-9 .
- SΓ‘ndor, Jozsef. Handbook of number theory II / Jozsef SΓ‘ndor, Borislav Crstici. - Dordrecht: Kluwer Academic, 2004. - P. 68. - ISBN 1-4020-2546-7 .
Links
- Weisstein, Eric W. Quasiamicable Pair (English) on the Wolfram MathWorld website.