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.

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, 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, 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

Offset: 0

Author

Lucas Colucci, Jun 25 2020

Keywords

• nonn
• easy
• base

Comments

The sequence can also be defined as the number of iterations of A089898 required to reach a fixed point or a cycle.

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).

It is likely, but not known, that this sequence is unbounded.

Examples

			17->16->14->10, which belongs to the cycle (2,3,...,10). Thus, a(17)=3.
44->25->18, which is a fixed point. Thus, a(44)=2.
		

Links

• Gabriel Bonuccelli, Lucas Colucci, and Edson de Faria, On the Erdős-Sloane and Shifted Sloane Persistence, arXiv:2009.01114 [math.NT], 2020.
• Samuel S. Wagstaff, Iterating the product of shifted digits, Fibonacci Quarterly 19.4 (1981): 340-347.

Crossrefs

Cf. A089898.

Programs

Maple

g:= n -> convert(map(`+`,convert(n,base,10),1),`*`):
f:= proc(n)
  local k, x, R;
  x:= n;
  R[x]:= 0;
  for k from 1 do
    x:= g(x);
    if assigned(R[x]) then return R[x] fi;
    R[x]:= k;
  od;
end proc:
map(f, [$0..100]); # Robert Israel, Jun 25 2020