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A166227 Let dsf(n) = n_1^{n_1}+n_2^{n_2}+n_3^{n_3} + n_m^{n_m}, where {n_1,n_2,n_3,...n_m} is the list of the digits of an integer n. dsf(793312220) = 388244100 and dsf(388244100) = 33554978, ..., dsf(387467199) = 793312220, ... in this way these 40 numbers make a loop for the function dsf.

%I A166227 #16 Aug 23 2023 18:27:39
%S A166227 793312220,388244100,33554978,405027808,34381363,16824237,17647707,
%T A166227 3341086,16824184,33601606,140025,3388,33554486,16830688,50424989,
%U A166227 791621836,405114593,387427281,35201810,16780376,18517643,17650825,17653671,1743552,830081,33554462,53476,873607,18470986,421845378,34381644,16824695,404294403,387421546,17651084,17650799,776537847,20121452,3396,387467199,793312220
%N A166227 Let dsf(n) = n_1^{n_1}+n_2^{n_2}+n_3^{n_3} + n_m^{n_m}, where {n_1,n_2,n_3,...n_m} is the list of the digits of an integer n. dsf(793312220) = 388244100 and dsf(388244100) = 33554978, ..., dsf(387467199) = 793312220, ... in this way these 40 numbers make a loop for the function dsf.
%C A166227 In fact there are only 8 loops among all the nonnegative integers for the "dsf" function that we defined. We have discovered this fact through calculations using Mathematica and general-purpose languages.
%H A166227 Ryohei Miyadera, <a href="http://library.wolfram.com/infocenter/MathSource/5686/">Curious Properties of an Iterative Process</a>, Mathsource, Wolfram Library Archive.
%H A166227 Shoei Takahashi, Unchone Lee, Hikaru Manabe, Aoi Murakami, Daisuke Minematsu, Kou Omori, and Ryohei Miyadera, <a href="https://arxiv.org/abs/2308.06691">Curious Properties of Iterative Sequences</a>, arXiv:2308.06691 [math.GM], 2023.
%F A166227 Let dsf(n) = n_1^{n_1}+n_2^{n_2}+n_3^{n_3} + n_m^{n_m}, where {n_1,n_2,n_3,...n_m} is the list of the digits of an integer n. By applying the function dsf to 793312220 we can get a loop of length 40.
%e A166227 This is an reiterative process that starts with 7793312220.
%t A166227 dsf[n_] := Block[{m = n, t}, t = IntegerDigits[m]; Sum[Max[1, t[[k]]]^t[[k]], {k, Length[t]}]]; NestList[dsf,7793312220,80]
%o A166227 (PARI) dsf(n) = my(d = digits(n)); sum(i=1, #d, d[i]^d[i]); \\ _Michel Marcus_, Apr 21 2014
%Y A166227 Cf. A165942, A166024, A166072, A166121.
%K A166227 base,nonn
%O A166227 1,1
%A A166227 _Ryohei Miyadera_, Takuma Nakaoka and Koichiro Nishimura, Oct 09 2009
%E A166227 More terms from _Michel Marcus_, Apr 21 2014