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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(791621579) = 776537851 and dsf(776537851) = 19300779, ..., dsf(824599) = 791621579, ... in this way these 11 numbers make a loop for the function dsf.
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%I #13 Aug 23 2023 18:27:34

%S 791621579,776537851,19300779,776488094,422669176,388384265,50381743,

%T 17604196,388337603,34424740,824599,791621579,776537851,19300779,

%U 776488094,422669176,388384265,50381743,17604196,388337603,34424740

%N 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(791621579) = 776537851 and dsf(776537851) = 19300779, ..., dsf(824599) = 791621579, ... in this way these 11 numbers make a loop for the function dsf.

%C 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 Ryohei Miyadera, <a href="http://library.wolfram.com/infocenter/MathSource/5686/">Curious Properties of an Iterative Process</a>,Mathsource, Wolfram Library Archive.

%H 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 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 791621579 we can get a loop of length 11.

%e This is an reiterative process that starts with 791621579.

%t dsf[n_] := Block[{m = n, t}, t = IntegerDigits[m]; Sum[Max[1, t[[k]]]^t[[k]], {k, Length[t]}]]; NestList[dsf,791621579,22]

%Y Cf. A165942, A166024, A166072.

%K base,nonn

%O 1,1

%A _Ryohei Miyadera_, Takuma Nakaoka and Koichiro Nishimura, Oct 07 2009