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.

0, 50, 70, 70, 70, 80, 1236, 40, 88, 10, 10, 50, 50, 60, 20, 50, 70, 50, 70, 10, 20, 50, 70, 50, 70, 80, 70, 10, 80, 90, 30, 60, 50, 70, 60, 90, 90, 40, 88, 90, 40, 20, 70, 60, 70, 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

Offset: 0

Author

M. F. Hasler, Apr 02 2025

Keywords

• nonn

Comments

The fixed points of A380873 are listed in A062237, except for 0.

The first two limit cycles that occur are both of length 5:

* C(88) = (88, 1664, 17144, 17112, 1214), reached for n = 8, 38, 83, 88, ... and

* C(18168) = (18168, 24384, 21768, 24672, 21672), reached for n = 188, 233, ...

Examples

			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.
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.
		

Crossrefs

Cf. A380873 (iterated function), A007953 (sum of digits), A007954 (product of digits), A062237 (nonzero fixed points of A380873), A380872 (trajectories under A380873).

Programs

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])