OFFSET
0,13
COMMENTS
a(n) = 0 iff n is already a palindrome (A002113).
Is it a theorem that a(n) always exists?
a(n) always exists. Proof: A palindrome can be reached by simply adding the initial digit until a palindrome with the same number of digits as the initial number is reached: If no palindrome is reached by then, this will yield a number with initial digit '1'. Thereafter, this procedure will yield the next larger palindrome - either not larger than 19...91 or, after 19...9 + 1 = 20...0, at 20...02. - M. F. Hasler, Apr 26 2014
REFERENCES
Eric Angelini, Posting to Sequence Fans Mailing List, Apr 20 2014
LINKS
David A. Corneth, Table of n, a(n) for n = 0..9999
David A. Corneth, PARI program
David A. Corneth, Steps taken from n as described in name to reach a palindrome
EXAMPLE
Examples for a(10) through a(23):
a(10) = 1 via 10 -> 11
a(11) = 0 via 11
a(12) = 3 via 12 -> 13 -> 16 -> 22
a(13) = 2 via 13 -> 16 -> 22
a(14) = 3 via 14 -> 15 -> 16 -> 22
a(15) = 2 via 15 -> 16 -> 22
a(16) = 1 via 16 -> 22
a(17) = 4 via 17 -> 18 -> 19 -> 20 -> 22
a(18) = 3 via 18 -> 19 -> 20 -> 22
a(19) = 2 via 19 -> 20 -> 22
a(20) = 1 via 20 -> 22
a(21) = 1 via 21 -> 22
a(22) = 0 via 22
a(23) = 3 via 23 -> 25 -> 30 -> 33
MATHEMATICA
A241173[n_] := Module[{c, nx},
If[n == IntegerReverse[n], Return[0]];
c = 1; nx = n;
While[ ! AnyTrue[nx = Flatten[nx + IntegerDigits[nx]], # == IntegerReverse[#] &], c++];
Return[c]];
Table[A241173[i], {i, 0, 100}] (* Robert Price, Mar 17 2019 *)
PROG
(PARI) a(n, m=0)={ if( m, my(d); for(i=1, #d=vecsort(digits(n), , 12), d[i] && if( m>1, a(n+d[i], m-1) /*&& !print1("/*", [n, d[i], m], "* /")*/, is_A002113(n+d[i])) && return(m)), is_A002113(n) || until(a(n, m++), ); m)} \\ Memoization should be implemented to improve performance which remains poor esp. for terms just above 10^k+1. - M. F. Hasler, Apr 26 2014
(PARI) \\ See Corneth link; faster than above. David A. Corneth, Mar 21 2019
CROSSREFS
KEYWORD
nonn,base
AUTHOR
N. J. A. Sloane, Apr 23 2014
EXTENSIONS
More terms from M. F. Hasler, Apr 26 2014
STATUS
approved