%N List of numbers in increasing order that belong to sociable cycles of length greater than 2 in which the sum of the cycle is divisible by 10.
%C Up to the known 1593 sociable number cycles, 96.1% of the sociable number cycles satisfy this condition (up to the first 10 sociable number cycles: 40%; up to the first 100 sociable number cycles: 77%; up to the first 500 sociable number cycles: 92%, and up to the first 1000 sociable number cycles: 94.9%). So the conjecture here is that as the number of sociable number cycles increases, the percentage of the sums of the sociable number cycles divisible by 10 approaches 100%. Notice that the sums of amicable pairs are similarly often divisible by 10, but are not included here (see A291422).
%D R. K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, 1994, pp. 62 - 63.
%D Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, Chappman and HALL/CRC, 2003, pp. 2747 - 2748.
%D Song Y. Yan, Perfect, Amicable and Sociable Numbers. A Computation Approach, World Scientific 1996, pp. 34 - 38.
%H Zoltan Galantai, <a href="/A292217/b292217.txt">Table of n, a(n) for n = 1..36</a>
%H Zoltan Galantai, <a href="/A292217/a292217_1.txt">List of known sociable number cycles</a> where the sums of the cycles is divisible by 10.
%H David Moews, <a href="http://djm.cc/sociable.txt">List of all known sociable number cycles</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SociableNumbers.html">Sociable Numbers</a>
%e The sum of 1264460, 1547860, 1727636 and 1305184 is divisible by ten, thus this sociable number cycle belongs to the sequence. On the other hand, the 12496, 14288, 15472, 14536, 14264 sociable number cycle does not qualify since its sum is 71506.
%Y Cf. A122726, A003416, A090615, A119478, A098007, A126016, A063990, A291422.
%A _Zoltan Galantai_, Sep 11 2017