cp's OEIS Frontend

A091048 a(n) = the number of steps needed to reach the final value of n via repeated interpretation of n as a base b+1 number where b is the largest digit of n.

%I A091048 #9 Jun 08 2025 16:15:42
%S A091048 0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,0,1,1,1,2,2,4,2,3,3,0,2,2,2,3,1,
%T A091048 3,4,3,4,0,2,2,2,3,3,1,2,1,4,0,3,3,3,4,2,4,3,2,5,0,3,4,4,2,3,2,5,5,5,
%U A091048 0,4,3,6,1,4,5,5,3,4,0,7,2,5,6,6,4,5,1,8,0,0,0,0,0,0,0,0,0,0,0,1
%N A091048 a(n) = the number of steps needed to reach the final value of n via repeated interpretation of n as a base b+1 number where b is the largest digit of n.
%C A091048 Any value of n with at least one digit 9 will not reduce further since 9+1 is 10 and n in base 10 is n. Also any single-digit number will likewise not reduce further. Such values of n therefore require 0 steps to reduce. Many terms reduce in very few steps and others take longer (88 for example, takes 8 steps). There is no maximum number of steps. See A091049 to see the first term requiring n steps. See A091047 to see the actual unchanging value reached for each value of n.
%H A091048 Robert Israel, <a href="/A091048/b091048.txt">Table of n, a(n) for n = 1..10000</a>
%H A091048 C. Seggelin, <a href="https://web.archive.org/web/20040109233146/http://www.plastereddragon.com/maths/bases.htm">Interesting Base Conversions</a>.
%e A091048 a(18)=4 because (1) 18 in base 9 is 17. (2) 17 in base 8 is 15. (3) 15 in base 6 is 11. (4) 11 in base 2 is 3. 3 does not reduce further because 3 in base 4 is 3. Thus 18 reduces to 3 in 4 steps.
%p A091048 f:= proc(n) option remember;
%p A091048    local L,b,i;
%p A091048    if n < 10 then return 0 fi;
%p A091048    L:= convert(n,base,10);
%p A091048    b:= max(L)+1;
%p A091048    if b = 10 then 0 else 1+procname(add(L[i]*b^(i-1),i=1..nops(L))) fi
%p A091048 end proc:
%p A091048 map(f, [$1..100]); # _Robert Israel_, Feb 19 2018
%Y A091048 Cf. A054055 (largest digit of n) A068505 (n as base b+1 number where b=largest digit of n) A091047 (a(n) = the final value of n reached through repeated interpretation of n as a base b+1 number where b is the largest digit of n) A091049 (a(n) = first term which reduces to an unchanging value in n steps via repeated interpretation of a(n) as a base b+1 number where b is the largest digit of a(n)).
%K A091048 base,nonn
%O A091048 1,15
%A A091048 _Chuck Seggelin_, Dec 15 2003