A261423 -id:A261423 - cp's OEIS frontend

A136687 Number of palindromes in the range [0,n] inclusive.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16

Offset: 0

N. J. A. Sloane, Apr 21 2008

Partial sums of A136522. [Typo fixed by Colin Barker, Apr 26 2015]

N. J. A. Sloane, Table of n, a(n) for n = 0..20202

Cf. A002113, A136522, A137180, A261423.

palQ[n_]:=Module[{idn=IntegerDigits[n]},idn==Reverse[idn]]; Accumulate[ Table[ If[palQ[n],1,0],{n,0,80}]] (* Harvey P. Dale, May 11 2014 *)

A136687(n)=inv_A002113(A261423(n)) \\ M. F. Hasler, Sep 09 2018

def A136687(n):
    l = len(s:=str(n))
    k = l+1>>1
    return n//10**(l-k)-(int(s[k-1::-1])>n%10**k)+10**(k-1+(l&1^1)) # Chai Wah Wu, Jul 24 2024

a(n) = inv_A002113(A261423(n)), where inv_A002113 is the inverse of A002113, i.e., it yields the index of a palindrome. - M. F. Hasler, Sep 10 2018

A261913 The palindromic order of n (defined in Comments).

Original entry on oeis.org

1, 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

Offset: 0

N. J. A. Sloane, Sep 10 2015

Order 1: palindromes (A002113);
Order 2: not order 1 but is the sum of two palindromes (A261907);
Order 3: not order 1 or 2, but n - previous_palindrome(n) (i.e., n - A261914(n)) gives a number of order 2 (A261910);
Order 4: not order 1, 2, or 3, but subtracting previous_palindrome(previous_palindrome(n)) gives a number of order 2 (A261911);
Order 5: not orders 1, 2, 3, or 4 (A261912).

N. J. A. Sloane, Table of n, a(n) for n = 0..20000

Cf. A002113, A261423, A261907, A261910, A261911, A261912, A261914.

Closely related to A261675. See also A088601.

a(n) = A088601(n). - R. J. Mathar, Feb 14 2023

A262037 Replace the second half of digits of n with the first half in reverse order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 77, 77, 77, 77, 77, 77, 77

Offset: 0

M. F. Hasler, Sep 08 2015

This is related to the "palindromic floor" and "palindromic ceiling" functions A261423 and A262038: a(n) = A261423(n) iff the first half of digits reversed does not exceed the second half of digits (without considering the middle digit in case of an odd number of digits), and a(n) = A262038(n) in the opposite case, i.e., first half of digits reversed is greater than or equal to second half of digits.
a(n) is either equal to the palindromic floor A261423(n) (e.g., a(1234) = 1221), or to the palindromic ceiling A262038(n) (= next larger palindrome, e.g., a(1324)=1331). In this sense it can be seen as a "palindromic round" function. However, it does not always yield the closest of the two (a(1900) = 1991 but 1881 would be closer to 1900). The sequence A262040 which has this property would better merit the name of "palindromic round function".
This simple function can be used to construct the next larger or next smaller palindrome, A261423 and A262038: indeed, if a(n) has the required property (less than or greater than n) then it is already the desired result, otherwise the result is given by a(n +- 10^k), where k is half the number of digits of n.

			a(31) = 33 since the second half ("1") gets replaced by the first half ("3").
a(314) = 313 since the second half ("4") is replaced by the first half ("3"), the middle "1" being untouched.
a(3141) = 3113 since the second half (41) is replaced by the first half (31), reversed (13).
a(31415) = 31413 as above, the middle 4 being left untouched.
a(314156) = 314413. This is the first instance in these examples where a(n) differs from A261423(n), which would yield 313313 here.

Dominic McCarty, Table of n, a(n) for n = 0..10000

Cf. A002113, A261423, A262038, A262039, A262040.

f[n_] := Block[{d = IntegerDigits@ n}, FromDigits[Take[d, Ceiling[Length[d]/2]]~Join~Reverse@ Take[d, Floor[Length[d]/2]]]]; Table[f@ n, {n, 0, 120}] (* Michael De Vlieger, Sep 09 2015 *)

a(n,d=digits(n),m=sum(k=1,#d\2,d[k]*10^(k-1)))=n+m-n%10^(#d\2)

def A262037(n):
    s = str(n); h = s[:(len(s)+1)//2]; return int(h + h[-1-len(s)%2::-1])
print([A262037(n) for n in range(77)]) # Michael S. Branicky, Sep 15 2022

A109326 Smallest positive number that requires n steps to be represented as a sum of palindromes using the greedy algorithm.

Original entry on oeis.org

1, 10, 21, 1022, 101023, 1000101024

Offset: 1

David Wasserman, Aug 11 2005

Index of first occurrence of n in A088601.
Presumably this sequence is unbounded. - N. J. A. Sloane, Aug 28 2015
The greedy algorithm means iteration of A261424 until a palindrome is reached. For n = 3, 4, ... we have a(n+1) = 10^L(n) + a(n) + 1 with L(n) = 2^(n-2) + 1 = length(a(n))*2 - 3 for n > 3. We have a(7) <= 10^17 + 1000101025, a(8) <= 10^33 + 10^17 + 1000101026, a(9) <= 10^65 + 10^33 + 10^17 + 1000101027, a(10) <= 10^129 + 10^65 + 10^33 + 10^17 + 1000101028, etc, with conjectured equality. - M. F. Hasler, Sep 08 2015, edited Sep 09 2018

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

Cf. A088601, A002113, A261422, A261423.

# uses functions in A088601
def afind(limit):
    record = 0
    for i in range(1, limit+1):
        steps = A088601(i)
        if steps > record: print(i, end=", "); record = steps
afind(10**6) # Michael S. Branicky, Jul 12 2021

a(n) = Sum_{0 <= k <= n-3} 10^(2^k+1) + n - 82, for n > 2 (conjectured). - M. F. Hasler, Sep 08 2015

Edited by N. J. A. Sloane, Aug 28 2015

A261914 Largest palindrome < n (or 0 if n=0).

Original entry on oeis.org

0, 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

Offset: 0

N. J. A. Sloane, Sep 10 2015

N. J. A. Sloane, Table of n, a(n) for n = 0..20000
Index entries for sequences related to palindromes

Similar to but of course different from A261423.

a261914 n = a261914_list !! n
a261914_list = f 0 a002113_list where
   f n ps@(p:ps'@(p':_)) = p : f (n + 1) (if n < p' then ps else ps')
-- Reinhard Zumkeller, Sep 16 2015

# Pal has list of palindromes; palfloor is A261423
palfloor:=proc(n) global Pal; local i;
for i from 1 to nops(Pal) do
   if Pal[i]=n then return(n); fi;
   if Pal[i]>n then return(Pal[i-1]); fi;
od:
end;
prevpal:=proc(n) global palfloor;
if n=0 then return(0);
elif member(n,Pal) then return(palfloor(n-1));
else return(palfloor(n)); fi; end;

lp[n_]:=Module[{k=n-1},While[!PalindromeQ[k],k--];k]; Join[{0},Array[lp,70]] (* Harvey P. Dale, Oct 17 2022 *)

a(n) = A261423(n-1) for all n>0. - M. F. Hasler, Sep 11 2015

A180458 Largest palindromic number <= n-th-prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 11, 11, 11, 22, 22, 22, 33, 33, 33, 44, 44, 55, 55, 66, 66, 66, 77, 77, 88, 88, 101, 101, 101, 101, 111, 121, 131, 131, 131, 141, 151, 151, 161, 161, 171, 171, 181, 191, 191, 191, 191, 202, 222, 222, 222, 232, 232, 232, 242, 252, 262, 262, 262, 272

Offset: 1

Giovanni Teofilatto, Sep 06 2010

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A261423.

# given lists Primes of primes and Palis of palindromes, with Palis[-1] > Primes[-1]
m:= 1; A:= 'A':
for n from 1 to nops(Primes) do
  while m < nops(Palis) and Palis[m+1] <= Primes[n] do m:= m+1 od:
  A[n]:= Palis[m]
od:
seq(A[i],i=1..nops(Primes)); # Robert Israel, Apr 26 2016

lpn[n_]:=Module[{k=0},While[!PalindromeQ[n-k],k++];n-k]; Table[lpn[n],{n,Prime[Range[60]]}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 18 2020 *)

Corrected by Robert Israel, Apr 26 2016

A262039 Nearest palindrome to n; in case of a tie choose the larger palindrome.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 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

Offset: 0

M. F. Hasler, Sep 08 2015

In analogy to the numerical "round" function, we "round up" to the next larger palindrome A262038(n) if it is at the same distance or closer, else we "round down" to the next smaller palindrome A261423(n). See A262040 for a variant where the next smaller palindrome is chosen in case of equal distance.

			a(10) = 11 since we round up if the next smaller palindrome (here 9) is at the same distance, both 9 and 11 are here at distance 1 from n = 10.
a(16) = 11 since |16 - 11| = 5 is smaller than |16 - 22| = 6.
a(17) = 22 since |17 - 22| = 5 is smaller than |17 - 11| = 6.
a(27) = 22 since |22 - 27| = 5 is smaller than |27 - 33| = 6.
a(28) = 33 since |33 - 28| = 5 is smaller than |22 - 28| = 6, and so on.
a(100) = 101 because we round up again in this case, where 99 and 101 both are at distance 1 from n = 100.

Cf. A002113, A261423, A262037, A262038, A262040.

palQ[n_] := Block[{d = IntegerDigits@ n}, d == Reverse@ d];
f[n_] := Block[{k = n}, While[Nand[palQ@ k, k > -1], k--]; k];
g[n_] := Block[{k = n}, While[! palQ@ k, k++]; k];
h[n_] := Block[{a = f@ n, b = g@ n}, Which[palQ@ n, n, (b - n) - (n - a) > 0, a, (b - n) - (n - a) <= 0, b]]; Table[h@ n, {n, 0, 73}] (* Michael De Vlieger, Sep 09 2015 *)

A262040 Nearest palindrome to n; in case of a tie choose the smaller palindrome.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 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

Offset: 0

M. F. Hasler, Sep 08 2015

In contrast to A262039, here we "round down" to the next smaller palindrome A261423(n) if it is at the same distance or closer, else we "round up" to the next larger palindrome A262038(n).

			a(10) = 9 since we round down if the next larger palindrome (here 11) is at the same distance, both 9 and 11 are here at distance 1 from n = 10.
a(16) = 11 since |16 - 11| = 5 is smaller than |16 - 22| = 6.
a(17) = 22 since |17 - 22| = 5 is smaller than |17 - 11| = 6.
a(27) = 22 since |22 - 27| = 5 is smaller than |27 - 33| = 6.
a(28) = 33 since |33 - 28| = 5 is smaller than |22 - 28| = 6, and so on.
a(100) = 99 because we round down in this case, where 99 and 101 both are at distance 1 from n = 100.

Cf. A002113, A261423, A262037, A262038, A262039.

palQ[n_] := Block[{d = IntegerDigits@ n}, d == Reverse@ d];
f[n_] := Block[{k = n}, While[Nand[palQ@ k, k > -1], k--]; k];
g[n_] := Block[{k = n}, While[! palQ@ k, k++]; k];
h[n_] := Block[{a = f@ n, b = g@ n}, Which[palQ@ n, n, (b - n) - (n - a) >= 0, a, (b - n) - (n - a) < 0, b]]; Table[h@ n, {n, 0, 73}] (* Michael De Vlieger, Sep 09 2015 *)

A262087 Largest palindrome p such that n-p is again a palindrome, or 0 if no such p exists.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 0, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 0, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 0, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 0, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 0, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 0, 77, 77, 77, 77, 77, 77

Offset: 0

M. F. Hasler, Sep 10 2015

Markus Sigg, Table of n, a(n) for n = 0..10000

Cf. A261423, A002113, A035137.

a(N,P=A261423(N))=until(N<2*P=A261423(P-1),is_A002113(N-P)&&return(P))

a(n) = 0 for n in {0} union A035137.

A262257 Minimal number of editing steps (delete, insert or substitute) to transform n in decimal representation into the largest palindrome <= n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 0, 1, 1, 2, 2, 2, 2, 2, 2

Offset: 0

Reinhard Zumkeller, Sep 16 2015

a(n) = Levenshtein distance between n and A261423(n);
0 <= a(n) <= A055642(n);
a(A002113(n)) = 0; a(m) = 0 iff A136522(m) = 1.

			.     n | A261423(n) | a(n)          n | A261423(n) | a(n)
.  -----+------------+-----      ------+------------+-------
.   100 |         99 |    3       1000 |        999 |    4
.   101 |        101 |    0       1001 |       1001 |    0
.   102 |        101 |    1       1002 |       1001 |    1
.   103 |        101 |    1       1003 |       1001 |    1
.   104 |        101 |    1       1004 |       1001 |    1
.   105 |        101 |    1       1005 |       1001 |    1
.   106 |        101 |    1       1006 |       1001 |    1
.   107 |        101 |    1       1007 |       1001 |    1
.   108 |        101 |    1       1008 |       1001 |    1
.   109 |        101 |    1       1009 |       1001 |    1
.   110 |        101 |    2       1010 |       1001 |    2
.   111 |        111 |    0       1011 |       1001 |    1
.   112 |        111 |    1       1012 |       1001 |    2
.   113 |        111 |    1       1013 |       1001 |    2
.   114 |        111 |    1       1014 |       1001 |    2
.   115 |        111 |    1       1015 |       1001 |    2
.   116 |        111 |    1       1016 |       1001 |    2
.   117 |        111 |    1       1017 |       1001 |    2
.   118 |        111 |    1       1018 |       1001 |    2
.   119 |        111 |    1       1019 |       1001 |    2
.   120 |        111 |    2       1020 |       1001 |    2
.   121 |        121 |    0       1021 |       1001 |    1
.   122 |        121 |    1       1022 |       1001 |    2
.   123 |        121 |    1       1023 |       1001 |    2
.   124 |        121 |    1       1024 |       1001 |    2
.   125 |        121 |    1       1025 |       1001 |    2 .

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Palindromic Number
WikiBooks: Algorithm Implementation, Levenshtein Distance
Wikipedia, Levenshtein Distance
Wikipedia, Palindromic number
Index entries for sequences related to palindromes

Cf. A261423, A002113, A136522, A055642.

import Data.Function (on); import Data.List (genericIndex)
a262257 n = genericIndex a262257_list n
a262257_list = zipWith (levenshtein `on` show) [0..] a261423_list where
   levenshtein us vs = last $ foldl transform [0..length us] vs where
     transform xs@(x:xs') c = scanl compute (x+1) (zip3 us xs xs') where
       compute z (c', x, y) = minimum [y+1, z+1, x + fromEnum (c' /= c)]

cp's OEIS Frontend

A136687 Number of palindromes in the range [0,n] inclusive.

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A261913 The palindromic order of n (defined in Comments).

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A262037 Replace the second half of digits of n with the first half in reverse order.

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A109326 Smallest positive number that requires n steps to be represented as a sum of palindromes using the greedy algorithm.

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Python

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A261914 Largest palindrome < n (or 0 if n=0).

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Haskell

Maple

Mathematica

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A180458 Largest palindromic number <= n-th-prime.

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Maple

Mathematica

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A262039 Nearest palindrome to n; in case of a tie choose the larger palindrome.

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Mathematica

A262040 Nearest palindrome to n; in case of a tie choose the smaller palindrome.

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