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A214301 Smallest limiting value of n under iteration of "Sum of its digits raised to its digits power" (A045512).

%I A214301 #20 May 24 2024 14:24:22
%S A214301 1,288,288,288,288,288,288,288,288,1,288,288,288,288,50119,3439,288,
%T A214301 288,288,288,288,288,288,288,288,288,288,288,288,288,288,288,288,288,
%U A214301 288,288,3439,288,288,288,288,288,288,288,288,288,288,288,288,288,50119
%N A214301 Smallest limiting value of n under iteration of "Sum of its digits raised to its digits power" (A045512).
%C A214301 Numbers that yield a fixed value after adding its digits raised to its digits power are called "Munchhausen Numbers" (A046253); i.e. 3435 = 3^3 + 4^4 + 3^3 + 5^5.
%C A214301 Since (9^9)*n < 10^n for n > 12; every initial value should eventually reach a "Munchhausen Number" or a cycle.
%C A214301 This sequence assigns to n that Munchhausen number or the lowest member of the cycle.
%C A214301 About 80% of the numbers reach the 288 cycle; 15% the 3439 cycle; 5% the 50119 cycle and less than 0.1% the fixed value 438579088; only few sporadic numbers reach 1 or 3435.
%C A214301 It has been proved that there are only four Munchhausen numbers (0, 1, 3435 and 438579088) with the convention of 0^0 = 0.
%C A214301 Open questions: are there any other cycles than those described here? What is the % of numbers < 10^n reaching a specific limiting value for n = 1, 2, 3, etc...?
%H A214301 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MuenchhausenNumber.html">Munchhausen Number</a>.
%t A214301 f[n_] := Module[{d = IntegerDigits[n]}, Sum[If[i == 0, 0, i^i], {i, d}]]; Table[s = NestWhileList[f, n, UnsameQ[##] &, All]; Min[Drop[s, Position[s, s[[-1]], 1, 1][[1, 1]]]], {n, 100}] (* _T. D. Noe_, Jul 12 2012 *)
%Y A214301 Cf. A046253, A045512.
%K A214301 base,nonn
%O A214301 1,2
%A A214301 _Sergio Pimentel_, Jul 11 2012