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%I #2 Mar 30 2012 18:49:09
%S 5,6,12,20,24,32,64,69,70,80,82,98,129,148,162,164,224,257,260,274,
%T 288,290,448,516,517,518,576,768
%N Numbers that belong to a cycle under the map k = Sum d_i 10^i -> f(k) = Sum d_i 2^i.
%C Obviously f(k) < k for k more than 3000. The Mathematica program below calculates f applied hundred times for every number up to 3000. After that the manual checking shows that the output is the exact list. There are three cycles. First: 70, 129, 518, 290, 517, 162. Second: 5, 32, 12, 6, 64, 80, 257, 164, 82, 260, 69, 576, 224, 24, 20. Third: 98, 768, 448, 288, 516.
%e f(70) = 2^7 + 2^0 = 129, f(129) = 2^1 + 2^2 + 2^9 = 518, f(518) = 2^5 + 2^1 + 2^8 = 290, f(290) = 2^2 + 2^9 + 2^0 = 517, f(517) = 2^5 + 2^1 + 2^7 = 162, f(162) = 2^1 + 2^6 + 2^2 = 70. That means that all the numbers 70, 129, 518, 290, 517, 162 belong to this sequence.
%t s2p[n_] := Plus @@ (2^IntegerDigits[n]); Union[Table[Nest[s2p, n, 100], {n, 3000}]]
%K base,fini,full,nonn
%O 1,1
%A _Tanya Khovanova_, Jan 05 2007
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