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Define dsf(n) = A045503(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 decimal digits of n. dsf(809265896) = 808491852 and dsf(808491852) = 437755524,...,dsf(792488396) = 809265896, so these 8 numbers make a loop for the function dsf.
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%I #19 Aug 23 2023 18:27:30

%S 809265896,808491852,437755524,1657004,873583,34381154,16780909,

%T 792488396,809265896,808491852,437755524,1657004,873583,34381154,

%U 16780909,792488396,809265896,808491852,437755524,1657004,873583

%N Define dsf(n) = A045503(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 decimal digits of n. dsf(809265896) = 808491852 and dsf(808491852) = 437755524,...,dsf(792488396) = 809265896, so these 8 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.

%C Periodic with period 8.

%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.

%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 a(n+1) = dsf(a(n)).

%t dsf[n_] := Block[{m = n, t}, t = IntegerDigits[m]; Sum[Max[1, t[[k]]]^t[[k]], {k, Length[t]}]]; NestList[dsf, 809265896,16]

%t LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 1},{809265896, 808491852, 437755524, 1657004, 873583, 34381154, 16780909, 792488396},24] (* _Ray Chandler_, Aug 25 2015 *)

%Y Cf. A165942, A166024.

%K nonn,base,easy

%O 1,1

%A _Ryohei Miyadera_, Satoshi Hashiba and Koichiro Nishimura, Oct 06 2009

%E Edited by _Charles R Greathouse IV_, Aug 02 2010

%E Extended by _Ray Chandler_, Aug 25 2015