A230107 Define a sequence by b(1)=n, b(k+1)=b(k)+(sum of digits of b(k)); a(n) is the number of steps needed to reach a term in A004207, or a(n) = -1 if the sequence never joins A004207.

0, 0, -1, 0, 52, -1, 11, 0, -1, 51, 50, -1, 49, 10, -1, 0, 48, -1, 9, 50, -1, 49, 0, -1, 47, 48, -1, 0, 8, -1, 49, 46, -1, 47, 48, -1, 45, 0, -1, 7, 46, -1, 47, 6, -1, 45, 44, -1, 0, 46, -1, 5, 5, -1, 45, 44, -1, 43, 4, -1, 4, 0, -1, 4, 44, -1, 43, 3, -1, 0

Offset: 0

Author

N. J. A. Sloane and Reinhard Zumkeller, Oct 15 2013; corrected Oct 20 2013

Keywords

• sign
• base

Comments

Looking at b(k) mod 9 shows that a(n) = -1 whenever n is a multiple of 3 (since then the b sequence is disjoint from A004207).

Conjecture: the b sequence, for any starting value n, will eventually merge with one of A000004 (the zero sequence), A004207, A016052 or A016096.

Examples

			For n=3, A016052 never meets A004207, so a(3) = -1.
For n=5, A007618 meets A004207 at the 53rd term, 620, so a(5) = 53.
		

Links

• Index entries for Colombian or self numbers and related sequences

Crossrefs

Cf. A004207, A016052, A007618, A006507, A016096, A062028, A230299.

Programs

Haskell

import Data.Maybe (fromMaybe)
a230107 = fromMaybe (-1) . f (10^5) 1 1 1 where
   f k i u j v | k <= 0    = Nothing
               | u < v     = f (k - 1) (i + 1) (a062028 u) j v
               | u > v     = f (k - 1) i u (j + 1) (a062028 v)
               | otherwise = Just j

Maple

read transforms; # to get digsum
M:=2000;
# f(s) returns the sequence k->k+digsum(k) starting at s
f:=proc(s) global M; option remember; local n,k,s1;
s1:=[s]; k:=s;
for n from 1 to M do  k:=k+digsum(k);
s1:=[op(s1),k]; od: end;
# g(s) returns (x,p), where x = first number in common between
# f(1) and f(s), and p is the position where it occurred.
# If f(1), f(s) are disjoint for M terms, returns (-1,-1)
S1:=convert(f(1),set):
g:=proc(s) global f,S1; local t1,p,S2,S3;
S2:=convert(f(s),set);
S3:= S1 intersect S2;
t1:=min(S3);
if (t1 = infinity) then RETURN(-1,-1); else
  member(t1,f(s),'p'); RETURN(t1,p-1); fi;
end;
[seq(g(n)[2],n=1..20)];