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%I #20 Sep 16 2017 00:27:08
%S 1,0,58,66,57,104,46,70,144,98,59,59,105,70,66,107,102,46,150,124,105,
%T 105,145,71,146,47,145,65,69,115,70,70,71,152,142,104,106,106,103,44,
%U 66,66,146,142,189,151,50,62,141,101,107,107,47,104,151,102,186,76
%N Let abc... be the decimal expansion of n. a(n) is the number of iterations of the map n -> f(n) needed to reach the last number of the cycle, where f(n) = a^a + b^b + c^c + ...
%C Additive persistence with powers of decimal digits: number of steps for "add digit(i) ^ digit(i)" operation to stabilize when started at n.
%C Or number of distinct values obtained by iterating n -> A045503(n).
%C We take 0^0 = 1.
%C It is conjectured that the trajectory for every number converges to a single number. The growth of a(n) is very slow; for example, a(457) = 211, a(10337) = 213, a(16669) = 214, ...
%H Michel Lagneau, <a href="/A226036/b226036.txt">Table of n, a(n) for n = 0..10000</a>
%e a(0) = 1 because 0 -> 0^0 = 1 with 1 iteration;
%e a(1) = 0 because 1 -> 1^1 => 0 iteration;
%e a(354) = 4 because:
%e 354 -> 3^3 + 5^5 + 4^4 = 3408;
%e 3408 -> 3^3 + 4^4 + 0^0 + 8^8 = 16777500;
%e 16777500 -> 1^1 + 6^6 + 7^7 + 7^7 + 7^7 + 5^5 + 0^0 + 0^0 = 2520413;
%e 2520413 -> 2^2 + 5^5 + 2^2 + 0^0 + 4^4 + 1^1 + 3^3 = 3418 and
%e 3418 is the last number of the cycle because 3418 -> 16777500 is already in the trajectory. We obtain 4 iterations: 354 -> 3408 -> 16777500 -> 2520413 -> 3418.
%p A000312:=proc(n)
%p if n = 0 then 1;
%p else add(d^d, d=convert(n, base, 10)) ;
%p end if;
%p end proc:
%p A226036:= proc(n)
%p local traj , c;
%p traj := n ;
%p c := [n] ;
%p while true do
%p traj := A000312(traj) ;
%p if member(traj, c) then
%p return nops(c)-1 ;
%p end if;
%p c := [op(c), traj] ;
%p end do:
%p end proc:
%p seq(A226036(n), n=0..100) ;
%t Unprotect[Power]; 0^0 = 1; Protect[Power]; f[n_] := (cnt++; id = IntegerDigits[n]; Total[id^id]); a[n_] := (cnt = 0; NestWhile[f, n, UnsameQ, All]; cnt-1); Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, May 24 2013 *)
%Y Cf. A000312, A031348, A031349, A045503, A133500, A225974.
%K nonn,base
%O 0,3
%A _Michel Lagneau_, May 24 2013
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