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%I #20 Aug 25 2015 14:05:11
%S 1,-1,3,-2,7,-6,24,12,4,-1,3,-2,7,-6,24,12,4,-1,3,-2,7,-6,24,12,4,-1,
%T 3,-2,7,-6,24,12,4,-1,3,-2,7,-6,24,12,4,-1,3,-2,7,-6,24,12,4,-1,3,-2,
%U 7,-6,24,12,4,-1,3,-2,7,-6,24,12,4,-1,3,-2,7,-6,24,12,4,-1,3,-2,7,-6,24,12,4,-1,3,-2,7,-6,24,12,4,-1,3,-2,7,-6
%N a(1)=1, a(n+1)=sigma(a(n))-2*a(n).
%C Taking any nonperfect number as initial value, does the map x->sigma(x)-2x lead to the cycle (-1,3,-2,7,-6,24,12,4) if during the iteration no perfect number is reached? Example: 124 -> -24 -> 108 -> 64 -> -1 -> 3 -> -2 -> 7 -> -6 -> 24 -> 12 -> 4 and the cycle (-1,3,-2,7,-6,24,12,4) is reached.
%C There appear to be lots of other cycles, for example the numbers in A005820 are cycles of length one. For longer cycles refer to the discussion in links. - _Hans Havermann_, Jul 21 2013
%H Peter Luschny, <a href="https://plus.google.com/u/0/102899428891193778097/posts/XbyqjcLoDjt">Open questions posed in OEIS #1</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 0, 0, 0, 0, 0, 1).
%F Periodic with period (-1, 3, -2, 7, -6, 24, 12, 4) of length 8.
%t NestList[DivisorSigma[1, #]-2#&, 1, 94] (* _Peter Luschny_, Jul 17 2013 *)
%t Join[{1},LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 1},{-1, 3, -2, 7, -6, 24, 12, 4},93]] (* _Ray Chandler_, Aug 25 2015 *)
%Y Cf. A146556, A005820, A069085, A113285.
%K sign
%O 1,3
%A _Benoit Cloitre_, Oct 02 2002