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A002113 - OEIS

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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). - <a href="/wiki/User:Hugo_Pfoertner">Hugo Pfoertner</a>, Aug 14 2015</div> <div class=sectline>This sequence is an additive basis of order at most 49, see Banks link. - <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Aug 23 2015</div> <div class=sectline>The order has been reduced from 49 to 3; see the Cilleruelo-Luca and Cilleruelo-Luca-Baxter links. - <a href="/wiki/User:Jonathan_Sondow">Jonathan Sondow</a>, Nov 27 2017</div> <div class=sectline>See <a href="/A262038" title="Least palindrome &gt;= n.">A262038</a> for the &quot;next palindrome&quot; and <a href="/A261423" title="Largest palindrome &lt;= n.">A261423</a> for the &quot;preceding palindrome&quot; functions. - <a href="/wiki/User:M._F._Hasler">M. F. Hasler</a>, Sep 09 2015</div> <div class=sectline>The number of palindromes with d digits is 10 if d = 1, and otherwise it is 9 * 10^(floor((d - 1)/2)). - <a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a>, Dec 06 2015</div> <div class=sectline>Sequence <a href="/A033665" title="Number of 'Reverse and Add' steps needed to reach a palindrome starting at n, or -1 if n never reaches a palindrome.">A033665</a> tells how many iterations of the Reverse-then-add function <a href="/A056964" title="a(n) = n + reversal of digits of n.">A056964</a> are needed to reach a palindrome; numbers for which this will never happen are Lychrel numbers (<a href="/A088753" title="Numbers n such that the Reverse and Add! trajectory of n (presumably) does not reach a palindrome (with the exception of n i...">A088753</a>) or rather Kin numbers (<a href="/A023108" title="Positive integers which apparently never result in a palindrome under repeated applications of the function A056964(x) = x +...">A023108</a>). - <a href="/wiki/User:M._F._Hasler">M. F. Hasler</a>, Apr 13 2019</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>Karl G. Kröber, &quot;Palindrome, Perioden und Chaoten: 66 Streifzüge durch die palindromischen Gefilde&quot; (1997, Deutsch-Taschenbücher; Bd. 99) ISBN 3-8171-1522-9.</div> <div class=sectline>Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 71.</div> <div class=sectline>N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).</div> <div class=sectline>N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>T. D. Noe, <a href="/A002113/b002113.txt">List of first 19999 palindromes: Table of n, a(n) for n = 1..19999</a></div> <div class=sectline>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.</div> <div class=sectline>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.</div> <div class=sectline>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.</div> <div class=sectline>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.</div> <div class=sectline>Patrick De Geest, <a href="http://www.worldofnumbers.com/">World of Numbers</a>.</div> <div class=sectline>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.</div> <div class=sectline>Phakhinkon Phunphayap and Prapanpong Pongsriiam, <a href="https://arxiv.org/abs/1803.00161">Reciprocal sum of palindromes</a>, arXiv:1803.00161 [math.CA], 2018.</div> <div class=sectline>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.</div> <div class=sectline>Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992</div> <div class=sectline>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.</div> <div class=sectline>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]</div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PalindromicNumber.html">Palindromic Number</a>.</div> <div class=sectline>Wikipedia, <a href="http://www.wikipedia.org/wiki/Palindromic_number">Palindromic number</a>.</div> <div class=sectline><a href="/index/Ab#basis_03">Index entries for sequences that are an additive basis</a>, order 3.</div> <div class=sectline><a href="/index/Pac#palindromes">Index entries for sequences related to palindromes</a></div> <div class=sectline><a href="/index/Cor#core">Index entries for &quot;core&quot; sequences</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline><a href="/A136522" title="a(n) = 1 if n is a palindrome, otherwise 0.">A136522</a>(a(n)) = 1.</div> <div class=sectline><a href="/A178788" title="Characteristic function of numbers having distinct digits in their decimal representation.">A178788</a>(a(n)) = 0 for n &gt; 9. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Jun 30 2010</div> <div class=sectline><a href="/A064834" title="If n (in base 10) is d_1 d_2 ... d_k then a(n) = Sum_{i = 1..[k/2] } |d_i - d_{k-i+1}|.">A064834</a>(a(n)) = 0. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Sep 18 2013</div> <div class=sectline>a(n+1) = <a href="/A262038" title="Least palindrome &gt;= n.">A262038</a>(a(n)+1). - <a href="/wiki/User:M._F._Hasler">M. F. Hasler</a>, Sep 09 2015</div> <div class=sectline>Sum_{n&gt;=2} 1/a(n) = <a href="/A118031" title="Decimal expansion of the sum of the reciprocals of the palindromic numbers A002113.">A118031</a>. - <a href="/wiki/User:Amiram_Eldar">Amiram Eldar</a>, Oct 17 2020</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline>read transforms; t0:=[]; for n from 0 to 2000 do if digrev(n) = n then t0:=[op(t0), n]; fi; od: t0;</div> <div class=sectline># Alternatively, to get all palindromes with &lt;= N digits in the list &quot;Res&quot;:</div> <div class=sectline>N:=5;</div> <div class=sectline>Res:= $0..9:</div> <div class=sectline>for d from 2 to N do</div> <div class=sectline> if d::even then</div> <div class=sectline> m:= d/2;</div> <div class=sectline> Res:= Res, seq(n*10^m + digrev(n), n=10^(m-1)..10^m-1);</div> <div class=sectline> else</div> <div class=sectline> m:= (d-1)/2;</div> <div class=sectline> Res:= Res, seq(seq(n*10^(m+1)+y*10^m+digrev(n), y=0..9), n=10^(m-1)..10^m-1);</div> <div class=sectline> fi</div> <div class=sectline>od: Res:=[Res]: # <a href="/wiki/User:Robert_Israel">Robert Israel</a>, Aug 10 2014</div> <div class=sectline># A variant: Gets all base-10 palindromes with exactly d digits, in the list &quot;Res&quot;</div> <div class=sectline>d:=4:</div> <div class=sectline>if d=1 then Res:= [$0..9]:</div> <div class=sectline>elif d::even then</div> <div class=sectline> m:= d/2:</div> <div class=sectline> Res:= [seq(n*10^m + digrev(n), n=10^(m-1)..10^m-1)]:</div> <div class=sectline>else</div> <div class=sectline> m:= (d-1)/2:</div> <div class=sectline> Res:= [seq(seq(n*10^(m+1)+y*10^m+digrev(n), y=0..9), n=10^(m-1)..10^m-1)]:</div> <div class=sectline>fi:</div> <div class=sectline>Res; # <a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a>, Oct 18 2015</div> <div class=sectline>isA002113 := proc(n)</div> <div class=sectline> simplify(digrev(n) = n) ;</div> <div class=sectline>end proc: # <a href="/wiki/User:R._J._Mathar">R. J. Mathar</a>, Sep 09 2015</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>palQ[n_Integer, base_Integer] := Module[{idn = IntegerDigits[n, base]}, idn == Reverse[idn]]; (* then to generate any base-b sequence for 1 &lt; b &lt; 37, replace the 10 in the following instruction with b: *) Select[Range[0, 1000], palQ[#, 10] &amp;]</div> <div class=sectline>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 (* <a href="/wiki/User:Arkadiusz_Wesolowski">Arkadiusz Wesolowski</a>, May 04 2012 *)</div> <div class=sectline>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 &gt; 2 *)</div> <div class=sectline>Array[nthPalindromeBase[#, 10] &amp;, 61, 0] (* please note that Schmidt uses a different, a more natural and intuitive offset, that of a(1) = 1. - <a href="/wiki/User:Robert_G._Wilson_v">Robert G. Wilson v</a>, Sep 22 2014 and modified Nov 28 2014 *)</div> <div class=sectline>Select[Range[10^3], PalindromeQ] (* <a href="/wiki/User:Michael_De_Vlieger">Michael De Vlieger</a>, Nov 27 2017 *)</div> <div class=sectline>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&gt;cn, fdpal, True, left=IntegerDigits[ FromDigits[left]+1]; pal=Join[left, Reverse[Take[left, Floor[len/2]]]]; FromDigits[pal]]]; NestList[nLP, 0, 100] (* <a href="/wiki/User:Harvey_P._Dale">Harvey P. Dale</a>, Dec 10 2024 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) is_<a href="/A002113" title="Palindromes in base 10.">A002113</a>(n)=Vecrev(n=digits(n))==n \\ <a href="/wiki/User:M._F._Hasler">M. F. Hasler</a>, Nov 17 2008, updated Apr 26 2014, Jun 19 2018</div> <div class=sectline>(PARI) is(n)=n=digits(n); for(i=1, #n\2, if(n[i]!=n[#n+1-i], return(0))); 1 \\ <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Jan 04 2013</div> <div class=sectline>(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])} \\ <a href="/wiki/User:David_A._Corneth">David A. Corneth</a>, Jun 06 2014</div> <div class=sectline>(PARI) \\ recursive--feed an element a(n) and it gives a(n+1)</div> <div class=sectline>nxt(n)=my(d=digits(n)); i=(#d+1)\2; while(i&amp;&amp;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]) \\ <a href="/wiki/User:David_A._Corneth">David A. Corneth</a>, Jun 06 2014</div> <div class=sectline>(PARI) \\ feed a(n), returns n.</div> <div class=sectline>inv(n)={my(d=digits(n)); q=ceil(#d/2); sum(i=1, q, 10^(q-i)*d[i])+10^floor(#d/2)} \\ <a href="/wiki/User:David_A._Corneth">David A. Corneth</a>, Jun 18 2014</div> <div class=sectline>(PARI) inv_<a href="/A002113" title="Palindromes in base 10.">A002113</a>(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. - <a href="/wiki/User:M._F._Hasler">M. F. Hasler</a>, Sep 09 2018</div> <div class=sectline>(PARI) <a href="/A002113" title="Palindromes in base 10.">A002113</a>(n, L=logint(n, 10))=(n-=L=10^max(L-(n&lt;11*10^(L-1)), 0))*L+fromdigits(Vecrev(digits(if(n&lt;L, n, n\10)))) \\ <a href="/wiki/User:M._F._Hasler">M. F. Hasler</a>, Sep 11 2018</div> <div class=sectline>(Python) # edited by <a href="/wiki/User:M._F._Hasler">M. F. Hasler</a>, Jun 19 2018</div> <div class=sectline>def <a href="/A002113" title="Palindromes in base 10.">A002113</a>_list(nMax):</div> <div class=sectline> mlist=[]</div> <div class=sectline> for n in range(nMax+1):</div> <div class=sectline> mstr=str(n)</div> <div class=sectline> if mstr==mstr[::-1]:</div> <div class=sectline> mlist.append(n)</div> <div class=sectline> return mlist # <a href="/wiki/User:Bill_McEachen">Bill McEachen</a>, Dec 17 2010</div> <div class=sectline>(Python)</div> <div class=sectline>from itertools import chain</div> <div class=sectline><a href="/A002113" title="Palindromes in base 10.">A002113</a> = 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)))) # <a href="/wiki/User:Chai_Wah_Wu">Chai Wah Wu</a>, Aug 09 2014</div> <div class=sectline>(Python)</div> <div class=sectline>from itertools import chain, count</div> <div class=sectline><a href="/A002113" title="Palindromes in base 10.">A002113</a> = chain(k for k in count(0) if str(k) == str(k)[::-1])</div> <div class=sectline>print([next(<a href="/A002113" title="Palindromes in base 10.">A002113</a>) for k in range(60)]) # <a href="/wiki/User:Jan_P._Hartkopf">Jan P. Hartkopf</a>, Apr 10 2021</div> <div class=sectline>(Python) is_<a href="/A002113" title="Palindromes in base 10.">A002113</a> = lambda n: (s:=str(n))[::-1]==s # <a href="/wiki/User:M._F._Hasler">M. F. Hasler</a>, May 23 2024</div> <div class=sectline>(Python)</div> <div class=sectline>from math import log10, floor</div> <div class=sectline>def <a href="/A002113" title="Palindromes in base 10.">A002113</a>(n):</div> <div class=sectline> if n &lt; 2: return 0</div> <div class=sectline> P = 10**floor(log10(n//2)); M = 11*P</div> <div class=sectline> s = str(n - (P if n &lt; M else M-P))</div> <div class=sectline> return int(s + s[-2 if n &lt; M else -1::-1]) # <a href="/wiki/User:M._F._Hasler">M. F. Hasler</a>, Jun 06 2024</div> <div class=sectline>(Haskell)</div> <div class=sectline> a002113 n = a002113_list !! (n-1)</div> <div class=sectline> a002113_list = filter ((== 1) . a136522) [1..] -- <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Oct 09 2011</div> <div class=sectline>(Haskell)</div> <div class=sectline> import Data.List.Ordered (union)</div> <div class=sectline> a002113_list = union a056524_list a056525_list -- <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Jul 29 2015, Dec 28 2011</div> <div class=sectline>(Magma) [n: n in [0..600] | Intseq(n, 10) eq Reverse(Intseq(n, 10))]; // <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Nov 03 2014</div> <div class=sectline>(SageMath)</div> <div class=sectline>[n for n in (0..515) if Word(n.digits()).is_palindrome()] # <a href="/wiki/User:Peter_Luschny">Peter Luschny</a>, Sep 13 2018</div> <div class=sectline>(GAP) Filtered([0..550], n-&gt;ListOfDigits(n)=Reversed(ListOfDigits(n))); # <a href="/wiki/User:Muniru_A_Asiru">Muniru A Asiru</a>, Mar 08 2019</div> <div class=sectline>(Scala) def palQ(n: Int, b: Int = 10): Boolean = n - Integer.parseInt(n.toString.reverse) == 0</div> <div class=sectline>(0 to 999).filter(palQ(_)) // <a href="/wiki/User:Alonso_del_Arte">Alonso del Arte</a>, Nov 10 2019</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Palindromes in bases 2 through 11: <a href="/A006995" title="Binary palindromes: numbers whose binary expansion is palindromic.">A006995</a> and <a href="/A057148" title="Palindromes only using 0 and 1 (i.e., base-2 palindromes).">A057148</a>, <a href="/A014190" title="Palindromes in base 3 (written in base 10).">A014190</a> and <a href="/A118594" title="Palindromes in base 3 (written in base 3).">A118594</a>, <a href="/A014192" title="Palindromes in base 4 (written in base 10).">A014192</a> and <a href="/A118595" title="Palindromes in base 4 (written in base 4).">A118595</a>, <a href="/A029952" title="Palindromic in base 5.">A029952</a> and <a href="/A118596" title="Palindromes in base 5 (written in base 5).">A118596</a>, <a href="/A029953" title="Palindromic in base 6.">A029953</a> and <a href="/A118597" title="Palindromes in base 6 (written in base 6).">A118597</a>, <a href="/A029954" title="Palindromic in base 7.">A029954</a> and <a href="/A118598" title="Palindromes in base 7 (written in base 7).">A118598</a>, <a href="/A029803" title="Numbers that are palindromic in base 8.">A029803</a> and <a href="/A118599" title="Palindromes in base 8 (written in base 8).">A118599</a>, <a href="/A029955" title="Palindromic in base 9.">A029955</a> and <a href="/A118600" title="Palindromes in base 9 (written in base 9).">A118600</a>, this sequence, <a href="/A029956" title="Numbers that are palindromic in base 11.">A029956</a>. Also <a href="/A262065" title="Numbers that are palindromes in base-60 representation.">A262065</a> (base 60), <a href="/A262069" title="Palindromes in base 10 that are also palindromes in base 60.">A262069</a> (subsequence).</div> <div class=sectline>Palindromic primes: <a href="/A002385" title="Palindromic primes: prime numbers whose decimal expansion is a palindrome.">A002385</a>. Palindromic nonprimes: <a href="/A032350" title="Palindromic nonprime numbers.">A032350</a>.</div> <div class=sectline>Palindromic-pi: <a href="/A136687" title="Number of palindromes in the range [0,n] inclusive.">A136687</a>.</div> <div class=sectline>Cf. <a href="/A029742" title="Nonpalindromic numbers.">A029742</a> (complement), <a href="/A086862" title="Differences between successive palindromes.">A086862</a> (first differences).</div> <div class=sectline>Palindromic floor function: <a href="/A261423" title="Largest palindrome &lt;= n.">A261423</a>, also <a href="/A261424" title="Difference between n and the largest palindrome &lt;= n.">A261424</a>. Palindromic ceiling: <a href="/A262038" title="Least palindrome &gt;= n.">A262038</a>.</div> <div class=sectline>Union of <a href="/A056524" title="Palindromes with even number of digits.">A056524</a> and <a href="/A056525" title="Palindromes with odd number of digits.">A056525</a>.</div> <div class=sectline>Cf. <a href="/A004086" title="Read n backwards (referred to as R(n) in many sequences).">A004086</a> (read n backwards), <a href="/A064834" title="If n (in base 10) is d_1 d_2 ... d_k then a(n) = Sum_{i = 1..[k/2] } |d_i - d_{k-i+1}|.">A064834</a>, <a href="/A118031" title="Decimal expansion of the sum of the reciprocals of the palindromic numbers A002113.">A118031</a>, <a href="/A136522" title="a(n) = 1 if n is a palindrome, otherwise 0.">A136522</a> (characteristic function), <a href="/A178788" title="Characteristic function of numbers having distinct digits in their decimal representation.">A178788</a>.</div> <div class=sectline>Ways to write n as a sum of three palindromes: <a href="/A261132" title="Number of ways to write n as the sum u+v+w of three palindromes (from A002113) with 0 &lt;= u &lt;= v &lt;= w.">A261132</a>, <a href="/A261422" title="Number of ordered triples (u,v,w) of palindromes such that u+v+w=n.">A261422</a>.</div> <div class=sectline>Minimal number of palindromes that add to n using greedy algorithm: <a href="/A088601" title="Number of steps to reach 0 when iterating A261424(x) = x - (the largest palindrome less than x), starting at n.">A088601</a>.</div> <div class=sectline>Minimal number of palindromes that add to n: <a href="/A261675" title="Minimal number of palindromes in base 10 that add to n.">A261675</a>.</div> <div class=sectline>Subsequence of <a href="/A061917" title="Either a palindrome or becomes a palindrome if trailing 0's are omitted.">A061917</a> and <a href="/A221221" title="Where powerbacks and powertrains coincide.">A221221</a>.</div> <div class=sectline>Subsequence: <a href="/A110745" title="a(n) is a number such that if odd positioned digits are deleted one gets n and if even positioned digits are deleted one get...">A110745</a>.</div> <div class=sectline>Sequence in context: <a href="/A297271" title="Numbers whose base-10 digits have equal down-variation and up-variation; see Comments.">A297271</a> <a href="/A110751" title="Numbers n such that n and its digital reversal have the same prime divisors.">A110751</a> <a href="/A147882" title="Positive integers k that are balanced, meaning that if k has d digits, then its initial ceiling(d/2) digits have the same su...">A147882</a> * <a href="/A227858" title="Numbers which start and end with the same digit in decimal.">A227858</a> <a href="/A335779" title="Curious numbers base 10.">A335779</a> <a href="/A240601" title="Recursive palindromes in base 10: palindromes n where each half of the digits of n is also a recursive palindrome.">A240601</a></div> <div class=sectline>Adjacent sequences: <a href="/A002110" title="Primorial numbers (first definition): product of first n primes. Sometimes written prime(n)#.">A002110</a> <a href="/A002111" title="Glaisher's G numbers.">A002111</a> <a href="/A002112" title="Glaisher's H numbers.">A002112</a> * <a href="/A002114" title="Glaisher's H' numbers.">A002114</a> <a href="/A002115" title="Generalized Euler numbers.">A002115</a> <a href="/A002116" title="Some special numbers.">A002116</a></div> </div> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <div class=sectline><span title="a sequence of nonnegative numbers">nonn</span>,<span title="dependent on base used for sequence">base</span>,<span title="it is very easy to produce terms of sequence">easy</span>,<span title="an exceptionally nice sequence">nice</span>,<span title="an important sequence">core</span>,<span title="edited within the last two weeks">changed</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a></div> </div> </div> <div class=section> <div class=sectname>STATUS</div> <div class=sectbody> <div class=sectline>approved</div> </div> </div> </div> <div class=space10></div> </div> </div></div> <p> <div class=footerpad></div> <div class=footer> <center> <div class=bottom> <div class=linksbar> <a href="/">Lookup</a> <a href="/wiki/Welcome"><font color="red">Welcome</font></a> <a href="/wiki/Main_Page"><font color="red">Wiki</font></a> <a href="/wiki/Special:RequestAccount">Register</a> <a href="/play.html">Music</a> <a href="/plot2.html">Plot 2</a> <a href="/demo1.html">Demos</a> <a href="/wiki/Index_to_OEIS">Index</a> <a href="/webcam">WebCam</a> <a href="/Submit.html">Contribute</a> <a href="/eishelp2.html">Format</a> <a href="/wiki/Style_Sheet">Style Sheet</a> <a href="/transforms.html">Transforms</a> <a href="/ol.html">Superseeker</a> <a href="/recent">Recents</a> </div> <div class=linksbar> <a href="/community.html">The OEIS Community</a> </div> <div class=linksbar> Maintained by <a href="http://oeisf.org">The OEIS Foundation Inc.</a> </div> <div class=dbinfo>Last modified December 11 02:47 EST 2024. Contains 378602 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>

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