A056525 -id:A056525 - cp's OEIS frontend

A002113 Palindromes in base 10.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 222, 232, 242, 252, 262, 272, 282, 292, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 404, 414, 424, 434, 444, 454, 464, 474, 484, 494, 505, 515

Offset: 1

N. J. A. Sloane

n is a palindrome (i.e., a(k) = n for some k) if and only if n = A004086(n). - Reinhard Zumkeller, Mar 10 2002
It seems that if n*reversal(n) is in the sequence then n = 3 or all digits of n are less than 3. - Farideh Firoozbakht, Nov 02 2014
The position of a palindrome within the sequence can be determined almost without calculation: If the palindrome has an even number of digits, prepend a 1 to the front half of the palindrome's digits. If the number of digits is odd, prepend the value of front digit + 1 to the digits from position 2 ... central digit. Examples: 98766789 = a(19876), 515 = a(61), 8206028 = a(9206), 9230329 = a(10230). - Hugo Pfoertner, Aug 14 2015
This sequence is an additive basis of order at most 49, see Banks link. - Charles R Greathouse IV, Aug 23 2015
The order has been reduced from 49 to 3; see the Cilleruelo-Luca and Cilleruelo-Luca-Baxter links. - Jonathan Sondow, Nov 27 2017
See A262038 for the "next palindrome" and A261423 for the "preceding palindrome" functions. - M. F. Hasler, Sep 09 2015
The number of palindromes with d digits is 10 if d = 1, and otherwise it is 9 * 10^(floor((d - 1)/2)). - N. J. A. Sloane, Dec 06 2015
Sequence A033665 tells how many iterations of the Reverse-then-add function A056964 are needed to reach a palindrome; numbers for which this will never happen are Lychrel numbers (A088753) or rather Kin numbers (A023108). - M. F. Hasler, Apr 13 2019
This sequence is an additive basis of order 3, see Cilleruelo, Luca, & Baxter and Sigg. - Charles R Greathouse IV, Apr 08 2025

Karl G. Kröber, "Palindrome, Perioden und Chaoten: 66 Streifzüge durch die palindromischen Gefilde" (1997, Deutsch-Taschenbücher; Bd. 99) ISBN 3-8171-1522-9.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 71.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 50-52.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 120.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, List of first 19999 palindromes: Table of n, a(n) for n = 1..19999
Hunki Baek, Sejeong Bang, Dongseok Kim, and Jaeun Lee, A bijection between aperiodic palindromes and connected circulant graphs, arXiv:1412.2426 [math.CO], 2014.
William D. Banks, Derrick N. Hart, and Mayumi Sakata, Almost all palindromes are composite, Math. Res. Lett., Vol. 11, No. 5-6 (2004), pp. 853-868.
William D. Banks, Every natural number is the sum of forty-nine palindromes, arXiv:1508.04721 [math.NT], 2015; Integers, 16 (2016), article A3.
Javier Cilleruelo, Florian Luca and Lewis Baxter, Every positive integer is a sum of three palindromes, Mathematics of Computation, Vol. 87, No. 314 (2018), pp. 3023-3055, arXiv preprint, arXiv:1602.06208 [math.NT], 2017.
Patrick De Geest, World of Numbers.
Kritkhajohn Onphaeng, Tammatada Khemaratchatakumthorn, Phakhinkon Napp Phunphayap, and Prapanpong Pongsriiam, Exact Formulas for the Number of Palindromes in Certain Arithmetic Progressions, Journal of Integer Sequences, Vol. 27 (2024), Article 24.4.8. See p. 2.
Phakhinkon Phunphayap and Prapanpong Pongsriiam, Reciprocal sum of palindromes, arXiv:1803.00161 [math.CA], 2018.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Prapanpong Pongsriiam and Kittipong Subwattanachai, Exact Formulas for the Number of Palindromes up to a Given Positive Integer, Intl. J. of Math. Comp. Sci. (2019) 14:1, 27-46.
E. A. Schmidt, Positive Integer Palindromes. [Cached copy at the Wayback Machine]
Markus Sigg, On a conjecture of John Hoffman regarding sums of palindromic numbers, arXiv preprint (2015). arXiv:1510.07507 [math.NT]
Eric Weisstein's World of Mathematics, Palindromic Number.
Wikipedia, Palindromic number.
Index entries for sequences that are an additive basis, order 3.
Index entries for sequences related to palindromes
Index entries for "core" sequences

Subsequence of A061917 and A221221.
A110745 is a subsequence.
Union of A056524 and A056525.
Palindromes in bases 2 through 11: A006995 and A057148, A014190 and A118594, A014192 and A118595, A029952 and A118596, A029953 and A118597, A029954 and A118598, A029803 and A118599, A029955 and A118600, this sequence, A029956. Also A262065 (base 60), A262069 (subsequence).
Palindromic primes: A002385. Palindromic nonprimes: A032350.
Palindromic-pi: A136687.
Cf. A029742 (complement), A086862 (first differences).
Palindromic floor function: A261423, also A261424. Palindromic ceiling: A262038.
Cf. A004086 (read n backwards), A064834, A118031, A136522 (characteristic function), A178788.
Ways to write n as a sum of three palindromes: A261132, A261422.
Minimal number of palindromes that add to n using greedy algorithm: A088601.
Minimal number of palindromes that add to n: A261675.

Filtered([0..550],n->ListOfDigits(n)=Reversed(ListOfDigits(n))); # Muniru A Asiru, Mar 08 2019

a002113 n = a002113_list !! (n-1)
  a002113_list = filter ((== 1) . a136522) [1..] -- Reinhard Zumkeller, Oct 09 2011

import Data.List.Ordered (union)
  a002113_list = union a056524_list a056525_list -- Reinhard Zumkeller, Jul 29 2015, Dec 28 2011

[n: n in [0..600] | Intseq(n, 10) eq Reverse(Intseq(n, 10))]; // Vincenzo Librandi, Nov 03 2014

read transforms; t0:=[]; for n from 0 to 2000 do if digrev(n) = n then t0:=[op(t0),n]; fi; od: t0;
# Alternatively, to get all palindromes with <= N digits in the list "Res":
N:=5;
Res:= $0..9:
for d from 2 to N do
  if d::even then
    m:= d/2;
    Res:= Res, seq(n*10^m + digrev(n),n=10^(m-1)..10^m-1);
  else
    m:= (d-1)/2;
    Res:= Res, seq(seq(n*10^(m+1)+y*10^m+digrev(n),y=0..9),n=10^(m-1)..10^m-1);
  fi
od: Res:=[Res]: # Robert Israel, Aug 10 2014
# A variant: Gets all base-10 palindromes with exactly d digits, in the list "Res"
d:=4:
if d=1 then Res:= [$0..9]:
elif d::even then
    m:= d/2:
    Res:= [seq(n*10^m + digrev(n), n=10^(m-1)..10^m-1)]:
else
    m:= (d-1)/2:
    Res:= [seq(seq(n*10^(m+1)+y*10^m+digrev(n), y=0..9), n=10^(m-1)..10^m-1)]:
fi:
Res; # N. J. A. Sloane, Oct 18 2015
isA002113 := proc(n)
    simplify(digrev(n) = n) ;
end proc: # R. J. Mathar, Sep 09 2015

palQ[n_Integer, base_Integer] := Module[{idn = IntegerDigits[n, base]}, idn == Reverse[idn]]; (* then to generate any base-b sequence for 1 < b < 37, replace the 10 in the following instruction with b: *) Select[Range[0, 1000], palQ[#, 10] &]
base10Pals = {0}; r = 2; Do[Do[AppendTo[base10Pals, n * 10^(IntegerLength[n] - 1) + FromDigits@Rest@Reverse@IntegerDigits[n]], {n, 10^(e - 1), 10^e - 1}]; Do[AppendTo[base10Pals, n * 10^IntegerLength[n] + FromDigits@Reverse@IntegerDigits[n]], {n, 10^(e - 1), 10^e - 1}], {e, r}]; base10Pals (* Arkadiusz Wesolowski, May 04 2012 *)
nthPalindromeBase[n_, b_] := Block[{q = n + 1 - b^Floor[Log[b, n + 1 - b^Floor[Log[b, n/b]]]], c = Sum[Floor[Floor[n/((b + 1) b^(k - 1) - 1)]/(Floor[n/((b + 1) b^(k - 1) - 1)] - 1/b)] - Floor[Floor[n/(2 b^k - 1)]/(Floor[n/(2 b^k - 1)] - 1/ b)], {k, Floor[Log[b, n]]}]}, Mod[q, b] (b + 1)^c * b^Floor[Log[b, q]] + Sum[Floor[Mod[q, b^(k + 1)]/b^k] b^(Floor[Log[b, q]] - k) (b^(2 k + c) + 1), {k, Floor[Log[b, q]]}]] (* after the work of Eric A. Schmidt, works for all integer bases b > 2 *)
Array[nthPalindromeBase[#, 10] &, 61, 0] (* please note that Schmidt uses a different, a more natural and intuitive offset, that of a(1) = 1. - Robert G. Wilson v, Sep 22 2014 and modified Nov 28 2014 *)
Select[Range[10^3], PalindromeQ] (* Michael De Vlieger, Nov 27 2017 *)
nLP[cn_Integer]:=Module[{s,len,half,left,pal,fdpal},s=IntegerDigits[cn]; len=Length[s]; half=Ceiling[len/2]; left=Take[s,half]; pal=Join[left,Reverse[ Take[left,Floor[len/2]]]]; fdpal=FromDigits[pal]; Which[cn==9,11,fdpal>cn,fdpal,True,left=IntegerDigits[ FromDigits[left]+1]; pal=Join[left,Reverse[Take[left,Floor[len/2]]]]; FromDigits[pal]]]; NestList[nLP,0,100] (* Harvey P. Dale, Dec 10 2024 *)

is_A002113(n)=Vecrev(n=digits(n))==n \\ M. F. Hasler, Nov 17 2008, updated Apr 26 2014, Jun 19 2018

is(n)=n=digits(n);for(i=1,#n\2,if(n[i]!=n[#n+1-i],return(0))); 1 \\ Charles R Greathouse IV, Jan 04 2013

a(n)={my(d,i,r);r=vector(#digits(n-10^(#digits(n\11)))+#digits(n\11));n=n-10^(#digits(n\11));d=digits(n);for(i=1,#d,r[i]=d[i];r[#r+1-i]=d[i]);sum(i=1,#r,10^(#r-i)*r[i])} \\ David A. Corneth, Jun 06 2014

\\ recursive--feed an element a(n) and it gives a(n+1)
nxt(n)=my(d=digits(n));i=(#d+1)\2;while(i&&d[i]==9,d[i]=0;d[#d+1-i]=0;i--);if(i,d[i]++;d[#d+1-i]=d[i],d=vector(#d+1);d[1]=d[#d]=1);sum(i=1,#d,10^(#d-i)*d[i]) \\ David A. Corneth, Jun 06 2014

\\ feed a(n), returns n.
inv(n)={my(d=digits(n));q=ceil(#d/2);sum(i=1,q,10^(q-i)*d[i])+10^floor(#d/2)} \\ David A. Corneth, Jun 18 2014

inv_A002113(P)={P\(P=10^(logint(P+!P,10)\/2))+P} \\ index n of palindrome P = a(n), much faster than above: no sum is needed. - M. F. Hasler, Sep 09 2018

A002113(n,L=logint(n,10))=(n-=L=10^max(L-(n<11*10^(L-1)),0))*L+fromdigits(Vecrev(digits(if(nM. F. Hasler, Sep 11 2018

# edited by M. F. Hasler, Jun 19 2018
def A002113_list(nMax):
  mlist=[]
  for n in range(nMax+1):
     mstr=str(n)
     if mstr==mstr[::-1]:
        mlist.append(n)
  return mlist # Bill McEachen, Dec 17 2010

from itertools import chain
A002113 = sorted(chain(map(lambda x:int(str(x)+str(x)[::-1]),range(1,10**3)),map(lambda x:int(str(x)+str(x)[-2::-1]), range(10**3)))) # Chai Wah Wu, Aug 09 2014

from itertools import chain, count
A002113 = chain(k for k in count(0) if str(k) == str(k)[::-1])
print([next(A002113) for k in range(60)]) # Jan P. Hartkopf, Apr 10 2021

is_A002113 = lambda n: (s:=str(n))[::-1]==s # M. F. Hasler, May 23 2024

from math import log10, floor
def A002113(n):
  if n < 2: return 0
  P = 10**floor(log10(n//2)); M = 11*P
  s = str(n - (P if n < M else M-P))
  return int(s + s[-2 if n < M else -1::-1]) # M. F. Hasler, Jun 06 2024

[n for n in (0..515) if Word(n.digits()).is_palindrome()] # Peter Luschny, Sep 13 2018

def palQ(n: Int, b: Int = 10): Boolean = n - Integer.parseInt(n.toString.reverse) == 0
(0 to 999).filter(palQ()) // _Alonso del Arte, Nov 10 2019

A136522(a(n)) = 1.
A178788(a(n)) = 0 for n > 9. - Reinhard Zumkeller, Jun 30 2010
A064834(a(n)) = 0. - Reinhard Zumkeller, Sep 18 2013
a(n+1) = A262038(a(n)+1). - M. F. Hasler, Sep 09 2015
Sum_{n>=2} 1/a(n) = A118031. - Amiram Eldar, Oct 17 2020
a(n) = (floor(d(n)/(c(n)*9 + 1)))*10^A055642(d(n)) + A004086(d(n)) where b(n, k) = ceiling(log((n + 1)/k)/log(10)), c(n) = b(n, 2) - b(n, 11) and d(n) = (n - A086573(b(n*(2 - c(n)), 2) - 1)/2 - 1). - Alan Michael Gómez Calderón, Mar 11 2025

A056524 Palindromes with even number of digits.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, 2002, 2112, 2222, 2332, 2442, 2552, 2662, 2772, 2882, 2992, 3003, 3113, 3223, 3333, 3443, 3553, 3663, 3773, 3883, 3993, 4004, 4114, 4224, 4334, 4444, 4554

Offset: 1

Henry Bottomley, Jun 16 2000

Concatenation of n with reverse of n (keeping leading zeros in the reverse).
A178788(a(n)) = 0 for n > 1. - Reinhard Zumkeller, Jun 30 2010
All of the terms are divisible by eleven. - James Burling, Aug 08 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..9999

Cf. A002113, A004086, A056525, A066492.
Cf. A020338, A055642, A178788.
Cf. A110745 (permutation).

a056524 n = a056524_list !! (n-1)
a056524_list = [read (ns ++ reverse ns) :: Integer |
                n <- [0..], let ns = show n]
-- Reinhard Zumkeller, Dec 28 2011

d[n_]:=IntegerDigits[n]; Table[FromDigits[Join[x=d[n],Reverse[x]]],{n,45}] (* Jayanta Basu, May 29 2013 *)
Select[Flatten[Table[Range[10^n,10^(n+1)-1],{n,1,3,2}]],PalindromeQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 22 2018 *)

def a(n): s = str(n); return int(s + s[::-1])
print([a(n) for n in range(1, 46)]) # Michael S. Branicky, Nov 02 2021

a(n) = n*10^A055642(n) + A004086(n).

a(n) = 11 * A066492(n).

A076609 Palindromic numbers with prime middle digit.

Original entry on oeis.org

2, 3, 5, 7, 121, 131, 151, 171, 222, 232, 252, 272, 323, 333, 353, 373, 424, 434, 454, 474, 525, 535, 555, 575, 626, 636, 656, 676, 727, 737, 757, 777, 828, 838, 858, 878, 929, 939, 959, 979, 10201, 10301, 10501, 10701, 11211, 11311, 11511, 11711, 12221

Offset: 1

Jani Melik, Oct 21 2002

There are no such with an even number of digits.

			a(12)=272=2^4*17 is palindromic number and its middle digit 7 is prime, a(13)=323=17*19 is palindromic number and its middle digit 2 is prime, a(14)=333=3^2*37 is palindromic number and its middle digit 3 is prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A032350, A056525.

ts_numprapal := proc(n) local ad,adr,midigit; ad := convert(n,base,10): adr := ListTools[Reverse](ad): if nops(ad) mod 2 = 0 then return 1; fi; midigit := op( (nops(ad)+1)/2,ad ): if (isprime( midigit )='true' and adr=ad) then return 0; else return 1; fi end: ts_num_pal := proc(i) if ts_numprapal(i) = 0 then return (i) fi end: anumpal := [seq(ts_num_pal(i), i=1..50000)]: anumpal;

pnpmdQ[n_]:=Module[{idn=IntegerDigits[n],len=IntegerLength[n]},OddQ[len] && PalindromeQ[n]&&PrimeQ[idn[[(len+1)/2]]]]; Select[Range[15000],pnpmdQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 08 2017 *)

A045571 Numbers that are palindromic, divisible by 11 and have an odd number of digits.

Original entry on oeis.org

121, 242, 363, 484, 616, 737, 858, 979, 10901, 11011, 12221, 13431, 14641, 15851, 17171, 18381, 19591, 20702, 21912, 22022, 23232, 24442, 25652, 26862, 28182, 29392, 30503, 31713, 32923, 33033, 34243, 35453, 36663, 37873, 39193, 40304

Offset: 1

Jeff Burch

All the palindromic numbers with an even number of digits are divisible by 11. The number of palindromic numbers with 2*k+1 digits that are divisible by 11 is (10^(k+1) + (-1)^k)/11, and their asymptotic relative density within the set of all palindromic numbers with an odd number of digits (A056525) is 1/11 (Schmidt, 1988). - Amiram Eldar, Jan 11 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000
Harvey Schmidt, Jr., Palindromes: Density and Divisibility, Mathematics Magazine, Vol. 61, No. 5 (1988), pp. 297-300.

Intersection of A056525 and A083852.

Select[11 * Range[4000], PalindromeQ[#] && OddQ[IntegerLength[#]] &] (* Amiram Eldar, Jan 11 2021 *)

A056142 Concatenate n, floor[n/10], floor[n/100] ... (but do not continue if floor[.]=0).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 222, 232, 242, 252, 262, 272, 282, 292, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 404, 414, 424, 434, 444, 454, 464, 474, 484, 494, 505, 515, 525, 535, 545, 555

Offset: 0

Henry Bottomley, Jun 15 2000

For 0 < n < 100, a(n) = A056525(n). If n has 3 digits, then a(n) is a palindrome if and only if n is. If n has 4 or 5 digits, then a(n) is a palindrome if and only if all digits of n are equal. - David Wasserman, May 23 2005

Conjecture: if n has 3 or more digits, a(n) is a palindrome only if all the digits of n are the same. It is easy to see that any palindrome can have at most 2 distinct digits: matching digits from the initial n in the concatenation matches each digit after the second with an earlier digit. - Franklin T. Adams-Watters, Sep 07 2006

Cf. A002113, A056143, A056525.

Table[FromDigits[Flatten[IntegerDigits/@DeleteCases[Floor[n/10^Range[ 0,5]],0]]],{n,0,60}] (* Harvey P. Dale, Jul 11 2020 *)

A076612 Palindromic numbers with nonprime middle digit.

Original entry on oeis.org

1, 4, 6, 8, 9, 101, 111, 141, 161, 181, 191, 202, 212, 242, 262, 282, 292, 303, 313, 343, 363, 383, 393, 404, 414, 444, 464, 484, 494, 505, 515, 545, 565, 585, 595, 606, 616, 646, 666, 686, 696, 707, 717, 747, 767, 787, 797, 808, 818, 848, 868, 888, 898, 909

Offset: 1

Jani Melik, Oct 21 2002

By definition, all terms have an odd number of digits. It is not surprising that the sequence of middle digits is 1, 4, 6, 8, 9, 0. - Harvey P. Dale, Jun 15 2024

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A002113, A056525, A076609.

ts_num_midpal := proc(n) local ad,adr,midigit; ad := convert(n,base,10): adr := ListTools[Reverse](ad): if nops(ad) mod 2 = 0 then return 1; fi; midigit := op( (nops(ad)+1)/2,ad ): if (isprime( midigit )='false' and adr=ad) then return 0; else return 1; fi end: ts_n_pal := proc(n) if ts_num_midpal(n) = 0 then return (i) fi end: anpal := [seq(ts_n_pal(i), i=1..50000)]: anpal;

Select[Range[1000],PalindromeQ[#]&&OddQ[IntegerLength[#]]&&!PrimeQ[IntegerDigits[#][[(IntegerLength[#]+1)/2]]]&] (* Harvey P. Dale, Jun 15 2024 *)

from itertools import chain, count, islice
def A076612_gen(): # generator of terms
    return chain((1,4,6,8,9),chain.from_iterable((int((s:=str(d))+e+s[::-1]) for d in range(10**l,10**(l+1)) for e in '014689') for l in count(0)))
A076612_list = list(islice(A076612_gen(),20)) # Chai Wah Wu, Jun 23 2022

A372149 Palindrome numbers consisting only of odd digits.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 33, 55, 77, 99, 111, 131, 151, 171, 191, 313, 333, 353, 373, 393, 515, 535, 555, 575, 595, 717, 737, 757, 777, 797, 919, 939, 959, 979, 999, 1111, 1331, 1551, 1771, 1991, 3113, 3333, 3553, 3773, 3993, 5115, 5335, 5555, 5775, 5995, 7117, 7337, 7557, 7777, 7997

Offset: 1

James S. DeArmon, Apr 20 2024

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from James S. DeArmon)
James S. DeArmon, Common LISP code to create A372149

Intersection of A002113 and A014261.

Cf. A056524, A056525.

Select[Range[8000], PalindromeQ[#] && Times@@Boole[OddQ[IntegerDigits[#]]] == 1 &] (* Stefano Spezia, Apr 30 2024 *)

from itertools import count, islice, product
def agen(): # generator of terms
    for d in count(1):
        for p in product("13579", repeat=d//2):
            left = "".join(p)
            for mid in [[""], "13579"][d&1]:
                yield int(left + mid + left[::-1])
print(list(islice(agen(), 60))) # Michael S. Branicky, Jun 01 2025

Missing 1551 inserted by Stefano Spezia, Apr 30 2024

cp's OEIS Frontend

A002113 Palindromes in base 10.

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A056524 Palindromes with even number of digits.

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A076609 Palindromic numbers with prime middle digit.

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A045571 Numbers that are palindromic, divisible by 11 and have an odd number of digits.

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A056142 Concatenate n, floor[n/10], floor[n/100] ... (but do not continue if floor[.]=0).

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A076612 Palindromic numbers with nonprime middle digit.

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