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A261423 Largest palindrome <= n. 86
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 77, 77 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Might be called the palindromic floor function.

Let P(n) = n with the second half of its digits replaced by the first half of the digits in reverse order. If P(n) <= n, then a(n) = P(n), else if n=10^k then a(n) = n-1, else a(n) = P(n-10^floor(d/2)), where d is the number of digits of n. - M. F. Hasler, Sep 08 2015

The largest differences of n - a(n) occur for n = m*R(2k) - 1, where 1 <= m <= 9 and R(k)=(10^k-1)/9. In this case, n - a(n) = 1.1*10^k - 1. - M. F. Hasler, Sep 05 2018

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Eric Weisstein's World of Mathematics, Palindromic Number

Wikipedia, Palindromic_number

Index entries for sequences related to palindromes

FORMULA

n - a(n) < 1.1*10^floor(d/2), where d = floor(log_10(n)) + 1 is the number of digits of n. - M. F. Hasler, Sep 05 2018

MAPLE

# 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;

MATHEMATICA

palQ[n_] := Block[{d = IntegerDigits@ n}, d == Reverse@ d]; Table[k = n;

While[Nand[palQ@ k, k > -1], k--]; k, {n, 0, 78}] (* Michael De Vlieger, Sep 09 2015 *)

PROG

(PARI) A261423(n, d=digits(n), m=sum(k=1, #d\2, d[k]*10^(k-1)))={if( n%10^(#d\2)<m, n==10^valuation(n, 10)&&return(n-1); d=digits(n-=10^(#d\2) /*#digits may decrease!*/); sum(k=1, #d\2, d[k]*10^(k-1)), m)+n-n%10^(#d\2)} \\ M. F. Hasler, Sep 08 2015, minor edit on Sep 05 2018

(Haskell)

a261423 n = a261423_list !! n

a261423_list = tail a261914_list  -- Reinhard Zumkeller, Sep 16 2015

CROSSREFS

Cf. A002113, A261424, A261914 (previous palindrome).

Cf. A262038.

Sequences related to palindromic floor and ceiling: A175298, A206913, A206914, A261423, A262038, and the large block of consecutive sequences beginning at A265509.

A262257(n) = Levenshtein distance between n and a(n). - Reinhard Zumkeller, Sep 16 2015

Sequence in context: A266325 A262087 A261914 * A262040 A122638 A297235

Adjacent sequences:  A261420 A261421 A261422 * A261424 A261425 A261426

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Aug 28 2015

STATUS

approved

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Last modified August 22 02:44 EDT 2019. Contains 326169 sequences. (Running on oeis4.)