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.

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, 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, 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

Offset: 1

Author

Chuck Seggelin, Dec 15 2003

Keywords

• base
• nonn

Comments

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.

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• C. Seggelin, Interesting Base Conversions.

Crossrefs

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)).

Programs

Maple

f:= proc(n) option remember;
   local L,b,i;
   if n < 10 then return 0 fi;
   L:= convert(n,base,10);
   b:= max(L)+1;
   if b = 10 then 0 else 1+procname(add(L[i]*b^(i-1),i=1..nops(L))) fi
end proc:
map(f, [$1..100]); # Robert Israel, Feb 19 2018