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%I #8 Aug 15 2023 07:44:53
%S 1583236420,16827317,18470991,792441996,1163132183,16823961,404291050,
%T 387424134,17601586,17697199,1163955211,387473430,18424896,421022094,
%U 387421016,17647705,2520668,16873662,17740759,389894501,808398820
%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(1583236420) =1682731 and dsf(1682731) = 18470991,...,dsf(388290999) = 1583236420,.. in this way this 97 numbers make a loop for the function dsf. In fact this is the longest loop for dsf function in the set of all nonnegative integers.
%C In fact there are only 8 loops in the whole nonnegative integers for the dsf-function that we defined. We have discovered this fact with the calculation by Mathematica and other general purpose languages. We have presented 6 loops to this On-Line Encyclopedia of Integer Sequences, and other two loops are in fact fixed points {1} and {3435}. It is easy to see that dsf(1) = 1 and dsf(3435) = 3^3+4^4+3^3+5^5=3435.
%H Ryohei Miyadera, <a href="http://library.wolfram.com/infocenter/MathSource/5686/">Curious Properties of an Iterative Process</a>, Mathsource, Wolfram Library Archive.
%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 1583236420 repeatedly we can get a loop of the length of 97.
%e This is an iterative process that starts with 1583236420.
%t dsf[n_] := Block[{m = n, t}, t = IntegerDigits[m]; Sum[Max[1, t[[k]]]^t[[k]], {k, Length[t]}]]; NestList[dsf,1583236420,194]
%Y Cf. A165942, A166024, A166072, A166121, A166227.
%K nonn,base
%O 1,1
%A _Ryohei Miyadera_, Oct 13 2009
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