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A292217 Conjectured 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.

%I A292217 #67 Mar 12 2025 10:40:26
%S A292217 1264460,1305184,1547860,1727636,4938136,5423384,5504056,5753864,
%T A292217 18656380,20522060,24289964,28158165,28630036,29902635,29971755,
%U A292217 30853845,81128632,91314968,91401368,96389032,209524210,230143790,231439570,246667790,498215416,506040584,510137384,583014136
%N A292217 Conjectured 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 A292217 This list is not known to be complete (564 might be a member). See A122726. - _N. J. A. Sloane_, Sep 17 2021
%C A292217 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 A292217 R. K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, 1994, pp. 62 - 63.
%D A292217 Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, Chappman and HALL/CRC, 2003, pp. 2747 - 2748.
%D A292217 Song Y. Yan, Perfect, Amicable and Sociable Numbers. A Computation Approach, World Scientific 1996, pp. 34 - 38.
%H A292217 Zoltan Galantai, <a href="/A292217/b292217.txt">Table of n, a(n) for n = 1..36</a>
%H A292217 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 A292217 Shyam Sunder Gupta, <a href="https://doi.org/10.1007/978-981-97-2465-9_6">Perfect, Multiply Perfect, and Sociable Numbers</a>, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 6, 185-207.
%H A292217 David Moews, <a href="http://djm.cc/sociable.txt">A list of currently known aliquot cycles of length greater than 2</a> [This list is not known to be complete.]
%H A292217 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SociableNumbers.html">Sociable Numbers</a>
%e A292217 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 A292217 Cf. A122726, A003416, A090615, A119478, A098007, A126016, A063990, A291422.
%K A292217 nonn,tabf
%O A292217 1,1
%A A292217 _Zoltan Galantai_, Sep 11 2017
%E A292217 Changed definition and added comment to point out that this sequence is only conjectural. - _N. J. A. Sloane_, Sep 17 2021