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A335824 Persistence of the 1-shifted Sloane's problem: number of iterations of "multiply together all the digits of a number (in base 10) shifted by +1" needed to reach a fixed point or a cycle.

%I A335824 #29 Jan 05 2025 19:51:41
%S A335824 2,1,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,0,2,1,1,1,2,3,1,2,4,5,2,1,1,2,3,
%T A335824 2,4,6,3,7,2,1,1,3,2,2,2,5,2,3,2,1,2,1,4,2,7,4,4,3,2,1,2,2,6,5,4,3,5,
%U A335824 7,2,1,3,4,3,2,4,5,6,5,2,1,1,5,7,3,3,7,5,2,2,1,2,2,2,2,2,2,2,2,2
%N A335824 Persistence of the 1-shifted Sloane's problem: number of iterations of "multiply together all the digits of a number (in base 10) shifted by +1" needed to reach a fixed point or a cycle.
%C A335824 The sequence can also be defined as the number of iterations of A089898 required to reach a fixed point or a cycle.
%C A335824 Wagstaff proved that a(n) is well-defined for every n; i.e., every number eventually converges to a fixed point or a cycle when iterating its digits shifted by 1. Moreover, the only fixed point is 18 and the only cycle is (2,3,...,10).
%C A335824 It is likely, but not known, that this sequence is unbounded.
%H A335824 Gabriel Bonuccelli, Lucas Colucci, and Edson de Faria, <a href="https://arxiv.org/abs/2009.01114">On the Erdős-Sloane and Shifted Sloane Persistence</a>, arXiv:2009.01114 [math.NT], 2020.
%H A335824 Samuel S. Wagstaff, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/19-4/wagstaff.pdf">Iterating the product of shifted digits</a>, Fibonacci Quarterly 19.4 (1981): 340-347.
%e A335824 17->16->14->10, which belongs to the cycle (2,3,...,10). Thus, a(17)=3.
%e A335824 44->25->18, which is a fixed point. Thus, a(44)=2.
%p A335824 g:= n -> convert(map(`+`,convert(n,base,10),1),`*`):
%p A335824 f:= proc(n)
%p A335824   local k, x, R;
%p A335824   x:= n;
%p A335824   R[x]:= 0;
%p A335824   for k from 1 do
%p A335824     x:= g(x);
%p A335824     if assigned(R[x]) then return R[x] fi;
%p A335824     R[x]:= k;
%p A335824   od;
%p A335824 end proc:
%p A335824 map(f, [$0..100]); # _Robert Israel_, Jun 25 2020
%Y A335824 Cf. A089898.
%K A335824 nonn,easy,base
%O A335824 0,1
%A A335824 _Lucas Colucci_, Jun 25 2020