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A380871 Limit of the trajectory of n under A380873: concatenate sum and product of digits, if it ends on a fixed point, otherwise the least element of the limit cycle.

%I A380871 #9 Apr 04 2025 22:36:35
%S A380871 0,50,70,70,70,80,1236,40,88,10,10,50,50,60,20,50,70,50,70,10,20,50,
%T A380871 70,50,70,80,70,10,80,90,30,60,50,70,60,90,90,40,88,90,40,20,70,60,70,
%U A380871 20,70,40,70,10,50,50,80,90,20,80,50,50,80,40,60,70,70,90,70,50,1236,70,70,70,70,50,10
%N A380871 Limit of the trajectory of n under A380873: concatenate sum and product of digits, if it ends on a fixed point, otherwise the least element of the limit cycle.
%C A380871 The fixed points of A380873 are listed in A062237, except for 0.
%C A380871 The first two limit cycles that occur are both of length 5:
%C A380871 * C(88) = (88, 1664, 17144, 17112, 1214), reached for n = 8, 38, 83, 88, ... and
%C A380871 * C(18168) = (18168, 24384, 21768, 24672, 21672), reached for n = 188, 233, ...
%e A380871 The trajectory of n = 1 under A380873 is: 1 -> concat(1, 1) = 11 -> concat(1+1, 1*1) = 21 -> concat(2+1, 2*1) = 32 -> concat(3+2, 3*2) = 56 -> concat(3+2, 3*2) = 1130 -> concat(1+1+3+0, 1*1*3*0) = 50 -> concat(5+0, 5*0) = 50, so a fixed point is reached, and a(1) = 50.
%e A380871 The trajectory of n = 8 under A380873 is: 8 -> concat(8, 8) = 88 -> concat(8+8, 8*8) = 1664 -> concat(1+6+6+4, 1*6*6*4) = 17144 -> concat(1+7+1+4+4, 1*7*1*4*4) = 17112 -> concat(1+7+1+1+2, 1*7*1*1*2) = 1214 -> concat(1+2+1+4, 1*2*1*4) = 88 -> 1664 etc.: here the limit 5-cycle C(88) = (88, 1664, 17144, 17112, 1214) is reached, so a(8) = min(C(88)) = 88.
%o A380871 (PARI) apply( {A380871(n)=for(i=0,1,my(S=[n]); while(!setsearch(S, n=A380873(n)), S=setunion(S,[n])); i&& n=S[1]);n}, [0..90])
%Y A380871 Cf. A380873 (iterated function), A007953 (sum of digits), A007954 (product of digits), A062237 (nonzero fixed points of A380873), A380872 (trajectories under A380873).
%K A380871 nonn
%O A380871 0,2
%A A380871 _M. F. Hasler_, Apr 02 2025