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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.
0
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
0,1
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.
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.
EXAMPLE
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.
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
CROSSREFS
Cf. A089898.
Sequence in context: A297237 A297234 A085857 * A297231 A056620 A316869
KEYWORD
nonn,easy,base
AUTHOR
Lucas Colucci, Jun 25 2020
STATUS
approved