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A002113 Palindromes in base 10.

%I A002113 M0484 N0178 #383 May 24 2025 16:19:53
%S A002113 0,1,2,3,4,5,6,7,8,9,11,22,33,44,55,66,77,88,99,101,111,121,131,141,
%T A002113 151,161,171,181,191,202,212,222,232,242,252,262,272,282,292,303,313,
%U A002113 323,333,343,353,363,373,383,393,404,414,424,434,444,454,464,474,484,494,505,515
%N A002113 Palindromes in base 10.
%C A002113 n is a palindrome (i.e., a(k) = n for some k) if and only if n = A004086(n). - _Reinhard Zumkeller_, Mar 10 2002
%C A002113 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
%C A002113 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
%C A002113 This sequence is an additive basis of order at most 49, see Banks link. - _Charles R Greathouse IV_, Aug 23 2015
%C A002113 The order has been reduced from 49 to 3; see the Cilleruelo-Luca and Cilleruelo-Luca-Baxter links. - _Jonathan Sondow_, Nov 27 2017
%C A002113 See A262038 for the "next palindrome" and A261423 for the "preceding palindrome" functions. - _M. F. Hasler_, Sep 09 2015
%C A002113 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
%C A002113 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
%C A002113 This sequence is an additive basis of order 3, see Cilleruelo, Luca, & Baxter and Sigg. - _Charles R Greathouse IV_, Apr 08 2025
%D A002113 Karl G. Kröber, "Palindrome, Perioden und Chaoten: 66 Streifzüge durch die palindromischen Gefilde" (1997, Deutsch-Taschenbücher; Bd. 99) ISBN 3-8171-1522-9.
%D A002113 Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 71.
%D A002113 Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 50-52.
%D A002113 Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 120.
%D A002113 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002113 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002113 T. D. Noe, <a href="/A002113/b002113.txt">List of first 19999 palindromes: Table of n, a(n) for n = 1..19999</a>
%H A002113 Hunki Baek, Sejeong Bang, Dongseok Kim, and Jaeun Lee, <a href="http://arxiv.org/abs/1412.2426">A bijection between aperiodic palindromes and connected circulant graphs</a>, arXiv:1412.2426 [math.CO], 2014.
%H A002113 William D. Banks, Derrick N. Hart, and Mayumi Sakata, <a href="http://dx.doi.org/10.4310/MRL.2004.v11.n6.a10">Almost all palindromes are composite</a>, Math. Res. Lett., Vol. 11, No. 5-6 (2004), pp. 853-868.
%H A002113 William D. Banks, <a href="http://arxiv.org/abs/1508.04721">Every natural number is the sum of forty-nine palindromes</a>, arXiv:1508.04721 [math.NT], 2015; <a href="https://www.emis.de/journals/INTEGERS/papers/q3/q3.Abstract.html">Integers</a>, 16 (2016), article A3.
%H A002113 Javier Cilleruelo, Florian Luca and Lewis Baxter, <a href="https://doi.org/10.1090/mcom/3221">Every positive integer is a sum of three palindromes</a>, Mathematics of Computation, Vol. 87, No. 314 (2018), pp. 3023-3055, <a href="http://arxiv.org/abs/1602.06208">arXiv preprint</a>, arXiv:1602.06208 [math.NT], 2017.
%H A002113 Patrick De Geest, <a href="http://www.worldofnumbers.com/">World of Numbers</a>.
%H A002113 Kritkhajohn Onphaeng, Tammatada Khemaratchatakumthorn, Phakhinkon Napp Phunphayap, and Prapanpong Pongsriiam, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Phunphayap/phun7.html">Exact Formulas for the Number of Palindromes in Certain Arithmetic Progressions</a>, Journal of Integer Sequences, Vol. 27 (2024), Article 24.4.8. See p. 2.
%H A002113 Phakhinkon Phunphayap and Prapanpong Pongsriiam, <a href="https://arxiv.org/abs/1803.00161">Reciprocal sum of palindromes</a>, arXiv:1803.00161 [math.CA], 2018.
%H A002113 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A002113 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A002113 Prapanpong Pongsriiam and Kittipong Subwattanachai, <a href="http://ijmcs.future-in-tech.net/14.1/R-Pongsriiam.pdf">Exact Formulas for the Number of Palindromes up to a Given Positive Integer</a>, Intl. J. of Math. Comp. Sci. (2019) 14:1, 27-46.
%H A002113 E. A. Schmidt, <a href="https://web.archive.org/web/20110126180310/http://eric-schmidt.com:80/eric/palindrome/index.html">Positive Integer Palindromes</a>. [Cached copy at the Wayback Machine]
%H A002113 Markus Sigg, <a href="https://arxiv.org/abs/1510.07507">On a conjecture of John Hoffman regarding sums of palindromic numbers</a>, arXiv preprint (2015). arXiv:1510.07507 [math.NT]
%H A002113 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PalindromicNumber.html">Palindromic Number</a>.
%H A002113 Wikipedia, <a href="http://www.wikipedia.org/wiki/Palindromic_number">Palindromic number</a>.
%H A002113 <a href="/index/Ab#basis_03">Index entries for sequences that are an additive basis</a>, order 3.
%H A002113 <a href="/index/Pac#palindromes">Index entries for sequences related to palindromes</a>
%H A002113 <a href="/index/Cor#core">Index entries for "core" sequences</a>
%F A002113 A136522(a(n)) = 1.
%F A002113 A178788(a(n)) = 0 for n > 9. - _Reinhard Zumkeller_, Jun 30 2010
%F A002113 A064834(a(n)) = 0. - _Reinhard Zumkeller_, Sep 18 2013
%F A002113 a(n+1) = A262038(a(n)+1). - _M. F. Hasler_, Sep 09 2015
%F A002113 Sum_{n>=2} 1/a(n) = A118031. - _Amiram Eldar_, Oct 17 2020
%F A002113 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
%p A002113 read transforms; t0:=[]; for n from 0 to 2000 do if digrev(n) = n then t0:=[op(t0),n]; fi; od: t0;
%p A002113 # Alternatively, to get all palindromes with <= N digits in the list "Res":
%p A002113 N:=5;
%p A002113 Res:= $0..9:
%p A002113 for d from 2 to N do
%p A002113   if d::even then
%p A002113     m:= d/2;
%p A002113     Res:= Res, seq(n*10^m + digrev(n),n=10^(m-1)..10^m-1);
%p A002113   else
%p A002113     m:= (d-1)/2;
%p A002113     Res:= Res, seq(seq(n*10^(m+1)+y*10^m+digrev(n),y=0..9),n=10^(m-1)..10^m-1);
%p A002113   fi
%p A002113 od: Res:=[Res]: # _Robert Israel_, Aug 10 2014
%p A002113 # A variant: Gets all base-10 palindromes with exactly d digits, in the list "Res"
%p A002113 d:=4:
%p A002113 if d=1 then Res:= [$0..9]:
%p A002113 elif d::even then
%p A002113     m:= d/2:
%p A002113     Res:= [seq(n*10^m + digrev(n), n=10^(m-1)..10^m-1)]:
%p A002113 else
%p A002113     m:= (d-1)/2:
%p A002113     Res:= [seq(seq(n*10^(m+1)+y*10^m+digrev(n), y=0..9), n=10^(m-1)..10^m-1)]:
%p A002113 fi:
%p A002113 Res; # _N. J. A. Sloane_, Oct 18 2015
%p A002113 isA002113 := proc(n)
%p A002113     simplify(digrev(n) = n) ;
%p A002113 end proc: # _R. J. Mathar_, Sep 09 2015
%t A002113 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] &]
%t A002113 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 *)
%t A002113 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 *)
%t A002113 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 *)
%t A002113 Select[Range[10^3], PalindromeQ] (* _Michael De Vlieger_, Nov 27 2017 *)
%t A002113 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 *)
%o A002113 (PARI) is_A002113(n)=Vecrev(n=digits(n))==n \\ _M. F. Hasler_, Nov 17 2008, updated Apr 26 2014, Jun 19 2018
%o A002113 (PARI) 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
%o A002113 (PARI) 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
%o A002113 (PARI) \\ recursive--feed an element a(n) and it gives a(n+1)
%o A002113 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
%o A002113 (PARI) \\ feed a(n), returns n.
%o A002113 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
%o A002113 (PARI) 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
%o A002113 (PARI) A002113(n,L=logint(n,10))=(n-=L=10^max(L-(n<11*10^(L-1)),0))*L+fromdigits(Vecrev(digits(if(n<L,n,n\10)))) \\ _M. F. Hasler_, Sep 11 2018
%o A002113 (Python) # edited by _M. F. Hasler_, Jun 19 2018
%o A002113 def A002113_list(nMax):
%o A002113   mlist=[]
%o A002113   for n in range(nMax+1):
%o A002113      mstr=str(n)
%o A002113      if mstr==mstr[::-1]:
%o A002113         mlist.append(n)
%o A002113   return mlist # _Bill McEachen_, Dec 17 2010
%o A002113 (Python)
%o A002113 from itertools import chain
%o A002113 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
%o A002113 (Python)
%o A002113 from itertools import chain, count
%o A002113 A002113 = chain(k for k in count(0) if str(k) == str(k)[::-1])
%o A002113 print([next(A002113) for k in range(60)]) # _Jan P. Hartkopf_, Apr 10 2021
%o A002113 (Python) is_A002113 = lambda n: (s:=str(n))[::-1]==s # _M. F. Hasler_, May 23 2024
%o A002113 (Python)
%o A002113 from math import log10, floor
%o A002113 def A002113(n):
%o A002113   if n < 2: return 0
%o A002113   P = 10**floor(log10(n//2)); M = 11*P
%o A002113   s = str(n - (P if n < M else M-P))
%o A002113   return int(s + s[-2 if n < M else -1::-1]) # _M. F. Hasler_, Jun 06 2024
%o A002113 (Haskell)
%o A002113   a002113 n = a002113_list !! (n-1)
%o A002113   a002113_list = filter ((== 1) . a136522) [1..] -- _Reinhard Zumkeller_, Oct 09 2011
%o A002113 (Haskell)
%o A002113   import Data.List.Ordered (union)
%o A002113   a002113_list = union a056524_list a056525_list -- _Reinhard Zumkeller_, Jul 29 2015, Dec 28 2011
%o A002113 (Magma) [n: n in [0..600] | Intseq(n, 10) eq Reverse(Intseq(n, 10))]; // _Vincenzo Librandi_, Nov 03 2014
%o A002113 (SageMath)
%o A002113 [n for n in (0..515) if Word(n.digits()).is_palindrome()] # _Peter Luschny_, Sep 13 2018
%o A002113 (GAP) Filtered([0..550],n->ListOfDigits(n)=Reversed(ListOfDigits(n))); # _Muniru A Asiru_, Mar 08 2019
%o A002113 (Scala) def palQ(n: Int, b: Int = 10): Boolean = n - Integer.parseInt(n.toString.reverse) == 0
%o A002113 (0 to 999).filter(palQ(_)) // _Alonso del Arte_, Nov 10 2019
%Y A002113 Subsequence of A061917 and A221221.
%Y A002113 A110745 is a subsequence.
%Y A002113 Union of A056524 and A056525.
%Y A002113 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).
%Y A002113 Palindromic primes: A002385. Palindromic nonprimes: A032350.
%Y A002113 Palindromic-pi: A136687.
%Y A002113 Cf. A029742 (complement), A086862 (first differences).
%Y A002113 Palindromic floor function: A261423, also A261424. Palindromic ceiling: A262038.
%Y A002113 Cf. A004086 (read n backwards), A064834, A118031, A136522 (characteristic function), A178788.
%Y A002113 Ways to write n as a sum of three palindromes: A261132, A261422.
%Y A002113 Minimal number of palindromes that add to n using greedy algorithm: A088601.
%Y A002113 Minimal number of palindromes that add to n: A261675.
%K A002113 nonn,base,easy,nice,core
%O A002113 1,3
%A A002113 _N. J. A. Sloane_