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A088601 Number of steps to reach 0 when iterating A261424(x) = x - (the largest palindrome less than x), starting at n. 9
1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 1, 2, 2, 2, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,10

COMMENTS

The sequence "minimum number of palindromes that sum up to n", A261675, coincides with this sequence up to a(301). But then a(302) = 3 since 302 = 292 + 9 + 1, whereas 302 = 111 + 191.

While it has been conjectured [proved by Cilleruelo & Luca, 2016 -Ed.] that every number can be represented as a sum of at most 3 palindromes, the terms of this sequence, which correspond to a greedy representation, can be larger than 3 (see A109326). For example, 1022 can be represented as 33 + 989, but a(1022) = 4, because the greedy decomposition gives 1022 = 1001 + 11 + 9 + 1. - Giovanni Resta, Aug 20 2015

Presumably this sequence is unbounded (compare A109326). - N. J. A. Sloane, Sep 02 2015

This sequence is unbounded. Let n(1) := 1. To construct n(j+1), take a natural number m with 10^m > n(j) and set n(j+1) := 10^(2m) + 1 + n(j). Then a(n(j)) = j. - Markus Sigg, Oct 26 2015

In A109326 an explicit formula for a smaller (conjectured sharp) upper bound was already given earlier. - M. F. Hasler, Sep 09 2018

LINKS

Giovanni Resta, Table of n, a(n) for n = 1..10000

William D. Banks, Every natural number is the sum of forty-nine palindromes, arXiv:1508.04721 [math.NT], 2015 and INTEGERS 16 (2016) A3

M. F. Hasler, Sum of palindromes, OEIS wiki, Sept. 2015

J. Cilleruelo, F. Luca, Every positive integer is a sum of three palindromes, arXiv:1602.06208 (2016).

Markus Sigg, On a conjecture of John Hoffman regarding sums of palindromic numbers, arXiv:1510.07507 [math.NT], 2015.

FORMULA

a(n) < log_2(log_10(n)) + 3. - M. F. Hasler, Sep 09 2018

EXAMPLE

a(10) = 2: f(10) = 10-9 = 1, f(1) = 1-1 = 0, two steps.

MAPLE

# From N. J. A. Sloane, Aug 28 2015

# P has list of palindromes

palfloor:=proc(n) global P; local i;

for i from 1 to nops(P) do

   if P[i]=n then return(n); fi;

   if P[i]>n then return(P[i-1]); fi;

od:

end;

GA:=proc(n) global P, palfloor; local a, i, k;

a:=1; k:=n;

for i from 1 to 30 do

  if k-palfloor(k)=0 then return(a);

  else k:=k-palfloor(k); a:=a+1; fi;

od; end;

[seq(GA(n), n=0..200)];

MATHEMATICA

Length@ NestWhileList[f, #, # > 0 &] & /@ Range@ 105 - 1 (* Michael De Vlieger, Oct 26 2015 *)

PROG

(PARI) ispal(n) = my(d=digits(n)); Vecrev(d)==d;

fp(n) = {while(!ispal(n), n--); n; }

a(n) = {nb = 0; while (n, n -= fp(n); nb++); nb; } \\ Michel Marcus, Aug 20 2015

/* The above fp() is extremely inefficient already for mid-sized numbers. The PARI function A261423 should be preferred.*/

(PARI) A088601(n)=for(i=1, oo, (n-=A261423(n))||return(i)) \\ M. F. Hasler, Sep 09 2018

CROSSREFS

Cf. A109326 gives index of first occurrence of n in this sequence ("greedy inverse").

Cf. A002113, A261132, A261422, A261423, A261424.

Sequence in context: A257317 A163376 A261913 * A261675 A028950 A094916

Adjacent sequences:  A088598 A088599 A088600 * A088602 A088603 A088604

KEYWORD

base,easy,nonn

AUTHOR

Amarnath Murthy, Oct 13 2003

EXTENSIONS

More terms from David Wasserman, Aug 11 2005

STATUS

approved

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Last modified September 25 18:47 EDT 2020. Contains 337344 sequences. (Running on oeis4.)