A056524 -id:A056524 - cp's OEIS frontend

A066492 a(n) = A056524(n)/11.

1, 2, 3, 4, 5, 6, 7, 8, 9, 91, 101, 111, 121, 131, 141, 151, 161, 171, 181, 182, 192, 202, 212, 222, 232, 242, 252, 262, 272, 273, 283, 293, 303, 313, 323, 333, 343, 353, 363, 364, 374, 384, 394, 404, 414, 424, 434, 444, 454, 455, 465, 475, 485, 495, 505, 515

Offset: 1

Vladeta Jovovic, Jan 09 2002

Cf. A056524, A067087, A071272.

Edited by N. J. A. Sloane at the suggestion of Artur Jasinski, Aug 24 2007

A002113 Palindromes in base 10.

Original entry on oeis.org

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

A020338 Doublets: base-10 representation is the juxtaposition of two identical strings.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 1010, 1111, 1212, 1313, 1414, 1515, 1616, 1717, 1818, 1919, 2020, 2121, 2222, 2323, 2424, 2525, 2626, 2727, 2828, 2929, 3030, 3131, 3232, 3333, 3434, 3535, 3636, 3737, 3838, 3939, 4040, 4141, 4242, 4343, 4444, 4545, 4646

Offset: 1

David W. Wilson

Reinhard Zumkeller, Table of n, a(n) for n = 1..9999
Eric Weisstein's World of Mathematics, Concatenation

Cf. concatenation of n and k*n: this sequence (k=1), A019550 (k=2), A019551 (k=3), A019552 (k=4), A019553 (k=5), A009440 (k=6), A009441 (k=7), A009470 (k=8), A009474 (k=9).

Cf. A004216, A056524.

Flat(List([1..2],d->List([10^(d-1)..10^d-1],n->(10^d+1)*n))); # Muniru A Asiru, Mar 31 2018

a020338 n = read (ns ++ ns) :: Integer  where ns = show n
-- Reinhard Zumkeller, Jun 07 2015

[Seqint(Intseq(n) cat Intseq(n)): n in [1..46]]; // Bruno Berselli, Mar 20 2013

seq(seq((10^d+1)*n, n = 10^(d-1)..10^d-1),d=1..3); # Robert Israel, Jan 02 2015

nxt[n_]:=Module[{idn=IntegerDigits[n], idn1=IntegerDigits[n]}, FromDigits[Join[idn, idn1]]];Array[nxt, 100] (* Vincenzo Librandi, Feb 04 2014 *)

a(n) = eval(Str(n,n)); \\ Michel Marcus, Sep 10 2015

[int(str(n)+str(n)) for n in range(1,47)] # Danny Rorabaugh, Oct 10 2015

a(n) = n*10^(A004216(n)+1) + n. - Reinhard Zumkeller, Aug 11 2007
G.f.: 11*x/(1-x)^2 - Sum_{d >= 1} 9*x^(10^d)*(100^d*x-10^d*x-100^d)/(1-x)^2. - Robert Israel, Jan 02 2015
a(n) = n || n. (Where "||" denotes "concatenate". See link "Concatenation".) - Halfdan Skjerning, Apr 01 2018

A056525 Palindromes with odd number of digits.

Original entry on oeis.org

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, 565

Offset: 1

Henry Bottomley, Jun 16 2000

Concatenation of all but final digits of n with reverse of n (keeping leading zeros in the reverse)

A178788(a(n)) = 0 for n > 9. [From Reinhard Zumkeller, Jun 30 2010]

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to final digits of numbers

Cf. A002113, A004086, A056524.

a056525 n = a056525_list !! (n-1)
a056525_list = [1..9] ++ [read (ns ++ [z] ++ reverse ns) |
                n <- [1..], let ns = show n, z <- "0123456789"]
-- Reinhard Zumkeller, Dec 28 2011

Join[Range[9],Table[FromDigits[Join[x=IntegerDigits[n],Reverse[Drop[x,-1]]]],{n,10,56}]] (* Jayanta Basu, May 29 2013 *)
Select[Flatten[Table[Range[10^n,10^(n+1)-1],{n,0,2,2}]],PalindromeQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 21 2020 *)

a(n) = floor[n/10]*10^A055642(n)+A004086(n)

A067087 Concatenation of n-th prime and its reverse.

Original entry on oeis.org

22, 33, 55, 77, 1111, 1331, 1771, 1991, 2332, 2992, 3113, 3773, 4114, 4334, 4774, 5335, 5995, 6116, 6776, 7117, 7337, 7997, 8338, 8998, 9779, 101101, 103301, 107701, 109901, 113311, 127721, 131131, 137731, 139931, 149941, 151151, 157751

Offset: 1

Amarnath Murthy, Jan 07 2002

Every term of the sequence is divisible by 11.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A056524.

Table[ ToExpression[ StringJoin[ ToString[Prime[n]], StringReverse[ ToString[ Prime[n]]]]], {n, 1, 40} ]
#*10^IntegerLength[#]+IntegerReverse[#]&/@Prime[Range[50]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 11 2017 *)

a(n) = { my(d=digits(prime(n))); fromdigits(concat(d,Vecrev(d))) } \\ Harry J. Smith, May 13 2010

a(n) = A056524(prime(n)). - Andrew Howroyd, Dec 07 2024

More terms from Robert G. Wilson v, Jan 09 2002

A110745 a(n) is a number such that if odd positioned digits are deleted one gets n and if even positioned digits are deleted one gets n reversed. Counting is from the LSB side. The position of LSB is one.

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

Amarnath Murthy, Aug 10 2005

Except for initial 0, rearrangement of numbers in A056524. They first differ at a(101) = 110011, while A056524(101) = 101101. If n has digits d_1 d_2 ... d_k, permute them to d_1 d_k d_2 d_{k-1} ... d_{floor(k/2)+1} and use that as index to A056524. - Franklin T. Adams-Watters, Jun 20 2006

			a(12) = 1221, deleting the LSB and the third digit 2 we get 12, deleting second and fourth digit we get 21.

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

Cf. A045918.

Cf. A056524, A031298.

import Data.List (transpose)
a110745 n = read (concat $ transpose [ns, reverse ns]) :: Integer
            where ns = show n
-- Reinhard Zumkeller, Feb 14 2015

More terms from Franklin T. Adams-Watters, Jun 20 2006

A240453 Greatest prime divisors of the palindromes with an even number of digits.

Original entry on oeis.org

11, 11, 11, 11, 11, 11, 11, 11, 11, 13, 101, 37, 11, 131, 47, 151, 23, 19, 181, 13, 11, 101, 53, 37, 29, 11, 11, 131, 17, 13, 283, 293, 101, 313, 19, 37, 11, 353, 11, 13, 17, 11, 197, 101, 23, 53, 31, 37, 227, 13, 31, 19, 97, 11, 101, 103, 11, 107, 109, 13

Offset: 1

Michel Lagneau, Apr 05 2014

Greatest prime divisor of A056524(n), or greatest prime divisor of the concatenation of a number n and reverse(n).

The palindromes with an even number of digits are composite numbers divisible by 11. There are many palindromic prime divisors, such as 11, 101, 131, 151, 181, 313, 353, ..., 30103, ...

			a(10) = 13 because the concatenation of 10 and 01 is 1001 = 7*11*13 where 13 is the greatest divisor of 1001.

Michel Lagneau, Table of n, a(n) for n = 1..10000

Cf. A056524, A006530, A066492, A240454.

with(numtheory):for n from 1 to 100 do:x:=convert(n,base,10):n1:=nops(x): s:=sum('x[i]*10^(n1-i)', 'i'=1..n1):y:=n*10^n1+s:z:=factorset(y):n2:=nops(z):d:=z[n2]:printf(`%d, `,d):od:

d[n_]:=IntegerDigits[n];Table[FactorInteger[FromDigits[Join[x=d[n],Reverse[x]]]][[-1,1]],{n,1,100}]
FactorInteger[#][[-1,1]]&/@Flatten[Table[Select[Range[10^n,10^(n+1)-1],PalindromeQ],{n,1,3,2}]] (* Harvey P. Dale, Dec 06 2021 *)

from sympy import primefactors
def a(n): s = str(n); return max(primefactors(int(s + s[::-1])))
print([a(n) for n in range(1, 61)]) # Michael S. Branicky, Nov 11 2021

a(n) = A006530(A056524(n)).

A240454 Smallest prime divisors of the palindromes with an even number of digits.

Original entry on oeis.org

11, 2, 3, 2, 5, 2, 7, 2, 3, 7, 11, 3, 11, 11, 3, 11, 7, 3, 11, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 11, 11, 3, 11, 11, 3, 7, 11, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 3, 5, 5, 3, 5, 5, 3, 5, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 7, 11, 3, 11, 11, 3, 11, 7, 3, 11, 2, 2, 2

Offset: 1

Michel Lagneau, Apr 05 2014

a(n) is the smallest prime divisor of A056524(n), or smallest prime divisor of the concatenation of a number n and reverse(n).

			a(10) = 7 because the concatenation of 10 and 01 is 1001 = 7*11*13 where 7 is the smallest divisor of 1001.

Michel Lagneau, Table of n, a(n) for n = 1..10000

Cf. A020639, A056524, A240453.

with(numtheory):for n from 1 to 100 do:x:=convert(n,base,10):n1:=nops(x): s:=sum('x[i]*10^(n1-i)', 'i'=1..n1):y:=n*10^n1+s:z:=factorset(y):n2:=nops(z):d:=z[1]:printf(`%d, `,d):od:

d[n_]:=IntegerDigits[n];Table[FactorInteger[FromDigits[Join[x=d[n],Reverse[x]]]][[1,1]],{n,1,100}]
Table[FactorInteger[#][[1,1]]&/@Select[Range[10^n,10^(n+1)-1],PalindromeQ],{n,1,3,2}]//Flatten (* Harvey P. Dale, Jul 19 2021 *)

from sympy import primefactors
def a(n): s = str(n); return min(primefactors(int(s + s[::-1])))
print([a(n) for n in range(1, 83)]) # Michael S. Branicky, Nov 02 2021

a(n) = A020639(A056524(n)).

A272232 Smallest k > 0 such that R_k//n//R_k is prime, where R_k is the repunit A002275(k) of length k and // denotes concatenation; or -1 if no such k exists.

Original entry on oeis.org

1, 9, -1, 1, 2, 1, 10, 3, 1, 1, 3, -1, 2, 3, 33, 1, 2, 1, 1, 21, 1, 2, -1, 1, 7, 48, 292, 4, 3, 1, 1, 2, 1, -1, 135, -1, 1, -1, 1, 34, 3, 3, 40, 2, -1, 1, 3, 1, 1, 32, 61, 1, 2, 1, 137, -1, 3, 1, 2, 42, 1, 14, 1, 262, 2, 22, -1, 3, 9, 2, 33, 73, 1, 3, 1, 2, 3, -1, 2, 2, 1

Offset: 0

Felix Fröhlich, Apr 23 2016

a(2) = -1 (see second comment in A258372).
a(n) = -1 if n > 0 is in A099814 (see fourth comment in A004022).
a(n) = -1 if n is of the form A000042(i)*10^j+A000042(i) for some j > i > 0, since the resulting number is divisible by A002275(k)//A000042(i).
a(n) = -1 if n is a term of A010785 with an even number of digits, since any number of the form 1..1d..d1..1 with an even number of digits d is divisible by 11.
a(n) = 1 if there exists an integer x such that n = (A002275(A004023(x))-A011557(x)-1)/10.
From Chai Wah Wu, Nov 07 2019: (Start)
a(n) = -1 if n has an even number of digits and is a multiple of 11. In particular, a(n) = -1 if n is a term of A056524.
a(n) = -1 if n = (10^k+1)(10^m-1)/9 for some m > 0, k >= 0.
(End)
a(140) > 20000. - Hans Havermann, May 21 2022

			a(0) = 1 since 101 is prime; a(1) refers to the prime 1111111111111111111.
a(124) = -1 because R_k//124//R_k is divisible by 125*10^k-1.

Hans Havermann, Table of n, a(n) for n = 0..139

Cf. A000042, A002275, A056524, A099814, A004023, A258372, A306861.

Table[SelectFirst[Range[10^4], PrimeQ@ FromDigits@ Flatten@ {#, IntegerDigits@ n, #} &@ Table[1, {#}] &], {n, 0, 91}] /. k_ /; MissingQ@ k -> 0 (* Michael De Vlieger, Apr 25 2016, Version 10.2 *)

a(n) = my(k=1); while(!ispseudoprime(eval(Str((10^k-1)/9, n, (10^k-1)/9))), k++); k

a(35)-a(80) from Giovanni Resta, May 01 2016

Escape clausae value changed to -1 by N. J. A. Sloane, May 17 2022

A333440 Numbers where each digit can be paired with a digit of the same value at another position so that two pairs can be nested but cannot otherwise overlap.

Original entry on oeis.org

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 1001, 1100, 1111, 1122, 1133, 1144, 1155, 1166, 1177, 1188, 1199, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, 2002, 2112, 2200, 2211, 2222, 2233, 2244, 2255, 2266, 2277, 2288, 2299, 2332, 2442, 2552, 2662, 2772, 2882

Offset: 1

Rémy Sigrist, Mar 21 2020

The term 0 is included by convention (we consider here that it has no digit).
Each term has an even number of digits, and each digit has an even number of occurrences; hence each term belongs to A283871.
This sequence has connections with A014486; in both sequences digits are balanced in some way.
All palindromes with even number of digits belong to this sequence.
For any n > 0, the concatenation of the first n terms of A333399 or the concatenation of the first n+1 terms of A333399 belong to this sequence.

			The digits of 5586557768 can be paired as follows:
    (55)(8(6(55)(77)6)8)
so 5586557768 belongs to this sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..4159 (terms < 10^6)

Cf. A014486, A056524, A283871, A333399.

is(n, base=10) = { my (u=0, s=0); while (n, my (d=n%base); if (u && s%base==d, u--; s\=base, u++; s=s*base+d); n\=base); u==0 }

11 | a(n). - David A. Corneth, Mar 07 2021

cp's OEIS Frontend

A066492 a(n) = A056524(n)/11.

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A002113 Palindromes in base 10.

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A020338 Doublets: base-10 representation is the juxtaposition of two identical strings.

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A056525 Palindromes with odd number of digits.

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A067087 Concatenation of n-th prime and its reverse.

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A110745 a(n) is a number such that if odd positioned digits are deleted one gets n and if even positioned digits are deleted one gets n reversed. Counting is from the LSB side. The position of LSB is one.

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