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%I #20 Sep 16 2017 00:36:35
%S 0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,5,2,21,2,1,1,1,3,
%T 2,3,6,8,19,6,1,1,2,5,21,3,4,12,17,4,1,1,3,2,3,5,4,15,4,3,1,1,7,2,4,
%U 14,16,4,16,4,1,1,5,6,3,2,5,11,13,15,1,1,5
%N Let abcd... be the decimal expansion of n. Number of iterations of the map n -> f(n) needed to reach a number < 10, where f(n) = a^b + c^d + ... which ends in an exponent or a base according as the number of digits is even or odd.
%C Inspired by the sequence A133501 (Number of steps for "powertrain" operation to converge when started at n). It is conjectured that the trajectory for each number converges to a single number < 10.
%C The conjecture is true, since f(x) < x trivially holds for x > 10^10 and I have verified that for every 10 <= x <= 10^10 there is a k such that f^(k)(x) < x, where f^(k) denotes f applied k times. Both the conventions 0^0 = 1 and 0^0 = 0 have been taken into account. - _Giovanni Resta_, May 28 2013
%H Michel Lagneau, <a href="/A226135/b226135.txt">Table of n, a(n) for n = 0..10000</a>
%e a(62) = 7 because:
%e 62 -> 6^2 = 36;
%e 36 -> 3^6 = 729;
%e 729 -> 7^2 + 9^1 = 58;
%e 58 -> 5^8 = 390625;
%e 390625 -> 3^9 + 0^6 + 2^5 = 19715;
%e 19715 -> 1^9 + 7^1 + 5^1 = 13;
%e 13 -> 1^3 = 1;
%e 62 -> 36 -> 729 -> 58 -> 390625 -> 19715 -> 13 -> 1 with 7 iterations.
%p A133501:= proc(n)
%p local a, i, n1, n2, t1, t2;
%p n1:=abs(n); n2:=sign(n); t1:=convert(n1, base, 10); t2:=nops(t1); a:=0;
%p for i from 0 to floor(t2/2)-1 do
%p a := a+t1[t2-2*i]^t1[t2-2*i-1];
%p od:
%p if t2 mod 2 = 1 then
%p a:=a+t1[1]; fi; RETURN(n2*a); end;
%p A226135:= proc(n)
%p local traj , c;
%p traj := n ;
%p c := [n] ;
%p while true do
%p traj := A133501(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(A226135(n), n=0..100) ;
%p # second Maple program:
%p f:= n-> `if`(n<10, n, `if`(is(length(n), odd), f(10*n+1),
%p iquo(irem(n, 100, 'r'), 10, 'h')^h+f(r))):
%p a:= n-> `if`(n<10, 0, 1+a(f(n))):
%p seq(a(n), n=0..100); # _Alois P. Heinz_, May 27 2013
%Y Cf. A000312, A031348, A031349, A045503, A133500, A225974.
%K nonn,base
%O 0,25
%A _Michel Lagneau_, May 27 2013
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