Betrothed numbers

Betrothed numbers or quasi-amicable numbers or reduced-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.[1][2] 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 (OEIS: A003502, OEIS: A003503) are: (48, 75), (140, 195), (1050, 1925), (1575, 1648), (2024, 2295), (5775, 6128).

The sequence of all betrothed numbers is

48, 75, 140, 195, 1050, 1575, 1648, 1925, 2024, 2295, 5775, 6128, 8892, 9504, 16587, 20735, 62744, 75495, 186615, 196664, 199760, 206504, 219975, 266000, 309135, 312620, ... (sequence A005276 in the OEIS)

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

Quasi-sociable numbers

Like 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.[3][4] They are generalizations of the concepts of betrothed numbers and quasiperfect numbers. The first quasi-sociable sequence, or quasi-sociable chain, was discovered by Mitchell Dickerman in 1997, it has length 8: (sequence A309227 in the OEIS)

  • 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

This is the only one known quasi-sociable chain, and the second quasi-sociable chain (if exists) must have all numbers >1012.

Quasi-Aliquot sequence

Like aliquot sequence, a quasi-aliquot sequence or reduced aliquot sequence is a sequence of nonnegative integers in which each term is the sum of the proper divisors except 1 of the previous term,[5][6] the term after 1 is 0 instead of −1 although is −1. If the sequence reaches the number 0, it ends, since 0 has infinitely many divisors. For example, the quasi-Aliquot sequence of 36 is 36 is 36, 54, 65, 18, 20, 21, 10, 7, 0.

Most quasi-aliquot sequences terminate at 0, all such sequences necessarily end with either a prime number followed by 0 or 1 followed by 0 (of course, only the sequence of 1 can reach 1, since is never 1).

It is widely believed that every quasi-aliquot sequence ends in one of these three ways: 0, a betrothed pair, or a quasi-sociable cycle (i.e. there are no quasiperfect numbers, and there are no numbers whose quasi-aliquot sequence is infinite, note that there are some numbers like 276, whose aliquot sequence may be infinite). It is also widely believed that almost all nonnegative numbers have quasi-aliquot sequence terminate at 0.

Notes
  1. Quasi-amicable numbers
  2. Tables of Aliquot Cycles
  3. Problem 23.- Divisors (V) Aliquot sequences, Sociable Numbers
  4. reduced aliquot cycle
  5. Quasi-aliquot Sequences?
  6. Autres processus itératifs (in French)

References
  • Hagis, Peter, jr; Lord, Graham (1977). "Quasi-Amicable Numbers". Math. Comput. 31 (138): 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 978-1-4020-4215-7. Zbl 1151.11300.
  • Sándor, Jozsef; Crstici, Borislav (2004). Handbook of Number Theory II. Dordrecht: Kluwer Academic. p. 68. ISBN 978-1-4020-2546-4. Zbl 1079.11001.
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