cp's OEIS Frontend

A165942 For a nonnegative integer n, define 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} lists digits of n. Then starting with a(1) = 3418, a(n+1) = dsf(a(n)).

%I A165942 #17 Aug 22 2023 07:15:30
%S A165942 3418,16777500,2520413,3418,16777500,2520413,3418,16777500,2520413,
%T A165942 3418,16777500,2520413,3418,16777500,2520413,3418,16777500,2520413,
%U A165942 3418,16777500,2520413,3418,16777500,2520413,3418,16777500,2520413,3418,16777500,2520413
%N A165942 For a nonnegative integer n, define 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} lists digits of n. Then starting with a(1) = 3418, a(n+1) = dsf(a(n)).
%C A165942 Period 3. In fact there are only 8 such loops among all the nonnegative integers for the "dsf" function that we defined.
%H A165942 Ryohei Miyadera, <a href="http://library.wolfram.com/infocenter/MathSource/5686/">Curious Properties of an Iterative Process</a>, Mathsource, Wolfram Library Archive.
%H A165942 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 A165942 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1).
%e A165942 a(2) = dsf(a(1)) = dsf(3418) = 3^3+4^4+1^1+8^8 = 16777500; a(3) = dsf(16777500) = 1^1+6^6+7^7+7^7+7^7+5^5+0^0+0^0 = 2520413; a(4) = dsf(2520413) = 2^2+5^5+2^2+0^0+4^4+1^1+3^3 = 3418.
%e A165942 This is an iterative process that starts with 3418.
%t A165942 dsf[n_] := Block[{m = n, t}, t = IntegerDigits[m]; Sum[Max[1, t[[k]]]^t[[k]], {k, Length[t]}]]; NestList[dsf, 3418, 6]
%t A165942 LinearRecurrence[{0, 0, 1},{3418, 16777500, 2520413},30] (* _Ray Chandler_, Aug 25 2015 *)
%Y A165942 dsf is A045503.
%K A165942 nonn,base,easy
%O A165942 1,1
%A A165942 _Ryohei Miyadera_, Daisuke Minematsu and Taishi Inoue, Oct 01 2009
%E A165942 Cross-reference from _Charles R Greathouse IV_, Nov 01 2009
%E A165942 Edited by _Charles R Greathouse IV_, Mar 18 2010
%E A165942 Extended by _Ray Chandler_, Aug 25 2015