mersenneforum.org Quasi-aliquot Sequences?
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 2019-07-14, 09:04 #1 sweety439     Nov 2016 23×97 Posts Quasi-aliquot Sequences? In mathematics, a "quasiperfect number" is a natural number n for which n is the sum of its non-trivial divisors (i.e., its divisors excluding 1 and n), similarly, we define "quasi-aliquot sequence", a quasi-aliquot sequence is a sequence of positive integers in which each term is the sum of the non-trivial divisors (i.e., its divisors excluding 1 and n) of the previous term. e.g. the quasi-aliquot sequence of 36 is 36, 54, 65, 18, 20, 21, 10, 7, 0. Does quasi-aliquot sequence always end with either 0 or quasi-amicable pair (betrothed pair, such as 48 and 75)? Last fiddled with by sweety439 on 2019-07-14 at 09:13
 2019-07-14, 10:12 #2 garambois     Oct 2011 24·3·7 Posts I know at least one more quasi 8-cycles : 0 1270824975 1 1467511664 2 1530808335 3 1579407344 4 1638031815 5 1727239544 6 1512587175 7 1215571544 8 1270824975 9 1467511664 10 1530808335 11 1579407344 ... ... I had done many tests a few years ago with different iterative processes : n --> s(n) + b s(n)=sigma(n)-n If b=0 : aliquot sequences If b=-1 : quasi-aliquot sequences For example, if b=38, it exists a 298-cycle !!! You can see on this page, but sorry, in french (s(n) is called sigma'(n) on this page) : http://www.aliquotes.com/autres_proc...iteratifs.html I had even tried to extend the sigma function to something other than integers, such as polynomials, or Gauss integers. But the problem is that there is always something of the conventional order in these last cases, see here : http://www.aliquotes.com/etendre_sigma.html
 2020-06-21, 21:45 #3 sweety439     Nov 2016 23·97 Posts Conjecture: there are no quasi n-cycles if n is odd, specially, there are no quasi-perfect numbers.

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