A002113 -id:A002113 - cp's OEIS frontend

A351717 Numbers whose maximal (or lazy) Lucas representation (A130311) is palindromic.

0, 2, 3, 5, 6, 10, 12, 14, 17, 20, 28, 30, 34, 36, 42, 46, 56, 61, 75, 77, 85, 92, 94, 101, 107, 115, 122, 128, 150, 166, 176, 198, 200, 211, 219, 233, 244, 246, 260, 271, 277, 288, 296, 310, 321, 345, 360, 396, 405, 441, 469, 484, 520, 522, 544, 562, 570, 588

Offset: 1

Amiram Eldar, Feb 17 2022

A001610(n) = Lucas(n+1) - 1 is a term for all n, since A001610(0) = 0 has the representation 0 and the representation of Lucas(n+1) - 1 is n 1's for n > 0.

			The first 10 terms are:
   n  a(n)  A130311(a(n))
   ----------------------
   1   0               0
   2   2               1
   3   3              11
   4   5             101
   5   6             111
   6  10            1111
   7  12           10101
   8  14           11011
   9  17           11111
  10  20          101101

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to palindromes.

Cf. A000032, A001610, A130311.

Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712.

lazy = Select[IntegerDigits[Range[6000], 2], SequenceCount[#, {0, 0}] == 0 &]; t = Total[# * Reverse @ LucasL[Range[0, Length[#] - 1]]] & /@ lazy; s = FromDigits /@ lazy[[TakeWhile[Flatten[FirstPosition[t, #] & /@ Range[Max[t]]], NumberQ]]]; Join[{0}, Position[s, _?PalindromeQ] // Flatten]

A046447 Apart from initial term, composite numbers with the property that the concatenation of their prime factors is a palindrome.

Original entry on oeis.org

1, 4, 8, 9, 16, 25, 27, 32, 39, 49, 64, 69, 81, 119, 121, 125, 128, 129, 159, 219, 243, 249, 256, 259, 329, 339, 343, 403, 429, 469, 507, 512, 625, 669, 679, 729, 795, 1024, 1207, 1309, 1329, 1331, 1533, 1547, 1587, 1589, 1703, 2023, 2048, 2097, 2187, 2319

Offset: 1

Patrick De Geest, Jul 15 1998

Prime factors considered with multiplicity. - Harvey P. Dale, Apr 20 2025

			81 is a term because 81 = 3 * 3 * 3 * 3 -> 3333 is palindromic.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A046448, A046449, A046450.

Cf. A027746, A018252, A136522, A002113.

a046447 n = a046447_list !! (n-1)
a046447_list = 1 : filter f [1..] where
   f x = length ps > 1 && ps' == reverse ps'
         where ps' = concatMap show ps; ps = a027746_row x
-- Reinhard Zumkeller, May 02 2014

concat[n_]:=Flatten[Table[IntegerDigits[First[n]],{Last[n]}]]; palQ[n_]:= Module[{x=Flatten[concat/@FactorInteger[n]]},x==Reverse[x]&&!PrimeQ[n]]; Select[Range[2500],palQ] (* Harvey P. Dale, May 24 2011 *)
cpfpQ[n_]:=PalindromeQ[FromDigits[Flatten[IntegerDigits/@Flatten[PadRight[{},#[[2]],#[[1]]]&/@FactorInteger[n]]]]]; Join[{1},Select[Range[2500],CompositeQ[ #]&&cpfpQ[#]&]] (* Harvey P. Dale, Apr 20 2025 *)

from sympy import factorint, isprime
A046447_list = [1]
for n in range(4, 10**6):
    if not isprime(n):
        s = ''.join([str(p)*e for p, e in sorted(factorint(n).items())])
        if s == s[::-1]:
            A046447_list.append(n) # Chai Wah Wu, Jan 03 2015

Definition slightly modified by Harvey P. Dale, Apr 20 2025

A083852 Decimal palindromes that are multiples of 11.

Original entry on oeis.org

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 121, 242, 363, 484, 616, 737, 858, 979, 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

Offset: 1

Reinhard Zumkeller, May 06 2003

A083850(a(n))>0; palindromes with even length are terms.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..1908

Intersection of A002113 and A008593.

Cf. A029951, A045638, A045639, A043040, A043041, A045642, A045643, A045644, A083851.

Select[Range[0, 5500, 11], PalindromeQ] (* Paolo Xausa, Jul 07 2025 *)

forstep(k=0, 10^5, 11, d=digits(k); d==Vecrev(d) && print1(k, ", ")) \\ Jeppe Stig Nielsen, May 08 2020

A228730 Lexicographically earliest sequence of distinct nonnegative integers such that the sum of two consecutive terms is a palindrome in base 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 16, 17, 27, 28, 38, 39, 49, 50, 51, 15, 7, 26, 18, 37, 29, 48, 40, 59, 42, 13, 9, 24, 20, 35, 31, 46, 53, 58, 8, 14, 19, 25, 30, 36, 41, 47, 52, 69, 32, 12, 10, 23, 21, 34, 43, 45, 54, 57, 44, 11, 22, 33, 55, 56, 65, 66, 75, 76, 85, 86, 95, 96

Offset: 0

Paul Tek, Aug 31 2013

From M. F. Hasler, Nov 09 2013: (Start)
At each step, choose the smallest number not occurring earlier and such that a(n+1)+a(n) are palindromes, for all n.
Conjectured to be a permutation of the nonnegative integers.
See A062932 where injectivity is replaced by monotonicity; the sequences differ from a(16)=15 on.
This is an "arithmetic" analog to sequences A228407 and A228410, where instead of the sum, the union of the digits of subsequent terms is considered. (End)

			a(1) + a(2) = 3.
a(2) + a(3) = 5.
a(3) + a(4) = 7.
a(4) + a(5) = 9.
a(5) + a(6) = 11.
a(6) + a(7) = 22.
a(7) + a(8) = 33.

Paul Tek, Table of n, a(n) for n = 0..10000 (corrected by Michel Marcus, Jan 19 2019)
Paul Tek, PERL program for this sequence

Cf. A002113, A055266.

Cf. A062932 (strictly increasing variant).

{a=0;u=0; for(n=1, 99, u+=1<A002113(a+k)&&(a=k)&&next(2)))} \\ M. F. Hasler, Nov 09 2013

Perl
```
See Link section.
```

from itertools import islice
def ispal(n): s = str(n); return s == s[::-1]
def agen(): # generator of terms
    aset, an, mink = {0}, 0, 1
    yield 0
    while True:
        k = mink
        while k in aset or not ispal(an + k): k += 1
        an = k; aset.add(an); yield an
        while mink in aset: mink += 1
print(list(islice(agen(), 70))) # Michael S. Branicky, Nov 07 2022

a(0)=0 added by M. F. Hasler, Nov 15 2013

A260254 Number of ways to write n as sum of two palindromes in decimal representation.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 5, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 4, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Reinhard Zumkeller, Jul 21 2015

a(A035137(n)) = 0; a(A260255(n)) > 0.

			.   n | a(n) |                                n | a(n) |
. ----+------+--------------------------    ----+------+--------------
.   0 |    1 |  0                            21 |    0 |  ./.
.   1 |    1 |  1                            22 |    2 |  22, 11+11
.   2 |    2 |  2, 1+1                       23 |    1 |  22+1
.   3 |    2 |  3, 2+1                       24 |    1 |  22+2
.   4 |    3 |  4, 3+1, 2+2                  25 |    1 |  22+3
.   5 |    3 |  5, 4+1, 3+2                  26 |    1 |  22+4
.   6 |    4 |  6, 5+1, 4+2, 3+3             27 |    1 |  22+5
.   7 |    4 |  7, 6+1, 5+2, 4+3             28 |    1 |  22+6
.   8 |    5 |  8, 7+1, 6+2, 5+3, 4+4        29 |    1 |  22+7
.   9 |    5 |  9, 8+1, 7+2, 6+3, 5+4        30 |    1 |  22+8
.  10 |    5 |  9+1, 8+2, 7+3, 6+4, 5+5      31 |    1 |  22+9
.  11 |    5 |  11, 9+2, 8+3, 7+4, 6+5       32 |    0 |  ./.
.  12 |    5 |  11+1, 9+3, 8+4, 7+5, 6+6     33 |    2 |  33, 22+11
.  13 |    4 |  11+2, 9+4, 8+5, 7+6          34 |    1 |  33+1
.  14 |    4 |  11+3, 9+5, 8+6, 7+7          35 |    1 |  33+2
.  15 |    3 |  11+4, 9+6, 8+7               36 |    1 |  33+3
.  16 |    3 |  11+5, 9+7, 8+8               37 |    1 |  33+4
.  17 |    2 |  11+6, 9+8                    38 |    1 |  33+5
.  18 |    2 |  11+7, 9+9                    39 |    1 |  33+6
.  19 |    1 |  11+8                         40 |    1 |  33+7
.  20 |    1 |  11+9                         41 |    1 |  33+8  .

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Hugo Pfoertner, Plot of first 10^6 terms
Hugo Pfoertner, Plot of first 3*10^7 terms

Cf. A002113, A136522, A035137, A260255.

a260254 n = sum $ map (a136522 . (n -)) $
               takeWhile (<= n `div` 2) a002113_list

a(n) = sum{A136522(n - A002113(k)): k = 1..floor(n/2)}.

A261675 Minimal number of palindromes in base 10 that add to n.

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, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 1, 2, 2, 2, 2

Offset: 0

N. J. A. Sloane, Sep 02 2015

This sequence coincides with A088601 for n <= 301, but differs at n=302.
Although A088601 and this sequence agree for a large number of terms, because of their importance they warrant separate entries.
Cilleruelo and Luca prove that a(n) <= 3 (in fact they prove this for any fixed base g>=5). - Danny Rorabaugh, Feb 26 2016

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Javier Cilleruelo and Florian Luca, Every positive integer is a sum of three palindromes, arXiv preprint arXiv:1602.06208 [math.NT], 2016.
William D. Banks, Every natural number is the sum of forty-nine palindromes, INTEGERS 17 (2016), 9 pp.
James Grime and Brady Haran, Every Number is the Sum of Three Palindromes (2018), Numberphile video

Cf. A002113, A035137, A088601, A260255, A261422.

ispal(n)=my(d=digits(n)); d==Vecrev(d);
a(n)=my(L=n\2,d,e); if(ispal(n), return(1)); d=[1]; while((e=fromdigits(d))<=L, if(ispal(n-e), return(2)); my(k=#d,i=(k+1)\2); while(i&&d[i]==9, d[i]=0; d[k+1-i]=0; i--); if(i, d[i]++; d[k+1-i]=d[i], d=vector(#d+1); d[1]=d[#d]=1)); 3; \\ Charles R Greathouse IV, Nov 12 2018

A332121 a(n) = 2*(10^(2n+1)-1)/9 - 10^n.

Original entry on oeis.org

1, 212, 22122, 2221222, 222212222, 22222122222, 2222221222222, 222222212222222, 22222222122222222, 2222222221222222222, 222222222212222222222, 22222222222122222222222, 2222222222221222222222222, 222222222222212222222222222, 22222222222222122222222222222, 2222222222222221222222222222222

Offset: 0

M. F. Hasler, Feb 09 2020

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275 (repunits R_n = (10^n-1)/9), A002276 (2*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332120 .. A332129 (variants with different middle digit 0, ..., 9).
Cf. A332131 .. A332191 (variants with different repeated digit 3, ..., 9).

A332121 := n -> 2*(10^(2*n+1)-1)/9-10^n;

Array[2 (10^(2 # + 1)-1)/9 - 10^# &, 15, 0]

apply( {A332121(n)=10^(n*2+1)\9*2-10^n}, [0..15])

def A332121(n): return 10**(n*2+1)//9*2-10**n

a(n) = 2*A138148(n) + 1*10^n = A002276(2n+1) - 10^n.
G.f.: (1 + 101*x - 300*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A043040 Numbers that are palindromic and divisible by 5.

Original entry on oeis.org

0, 5, 55, 505, 515, 525, 535, 545, 555, 565, 575, 585, 595, 5005, 5115, 5225, 5335, 5445, 5555, 5665, 5775, 5885, 5995, 50005, 50105, 50205, 50305, 50405, 50505, 50605, 50705, 50805, 50905, 51015, 51115, 51215, 51315, 51415, 51515, 51615, 51715, 51815, 51915

Offset: 1

Clark Kimberling

Or, 0 and numbers that are palindromic and begin with 5.

Jaroslav Krizek, Table of n, a(n) for n = 1..2223

Intersection of A002113 and A008587.

Cf. A029951, A045638, A045639, A045641, A045642, A045643, A045644, A083852.

palQ[n_Integer, base_Integer] := Module[{idn = IntegerDigits[n, base]}, idn == Reverse[idn]];
Join[{0},Select[Range[5,100000,5], IntegerDigits[#][[1]] == 5 && palQ[#, 10] &]] (* T. D. Noe, Mar 12 2013 *)
Select[5*Range[0,11000],IntegerDigits[#]==Reverse[IntegerDigits[#]]&] (* Harvey P. Dale, Nov 29 2015 *)

Edited to include a(1) = 0 by Paolo Xausa, Jul 07 2025

A064834 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}|.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Oct 25 2001

Might be called the Palindromic Deviation (or PD(n)) of n, since it measures how far n is from being a palindrome. - W. W. Kokko, Mar 13 2013

a(A002113(n)) = 0; a(A029742(n)) > 0; A136522(n) = A000007(a(n)). - Reinhard Zumkeller, Sep 18 2013

			a(456) = | 4 - 6 | = 2, a(4567) = | 4 - 7 | + | 5 - 6 | = 4.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Index entries for sequences related to palindromes

Cf. A037904, A040163, A064844.

Cf. A031298, A037888.

a064834 n = sum $ take (length nds `div` 2) $
                  map abs $ zipWith (-) nds $ reverse nds
   where nds = a031298_row n
-- Reinhard Zumkeller, Sep 18 2013

f:=proc(n)
local t1,t2,i;
t1:=convert(n,base,10);
t2:=nops(t1);
add( abs(t1[i]-t1[t2+1-i]),i=1..floor(t2/2) );
end;
[seq(f(n),n=0..120)]; # N. J. A. Sloane, Mar 24 2013

f[n_] := (k = IntegerDigits[n]; l = Length[k]; Sum[ Abs[ k[[i]] - k[[l - i + 1]]], {i, 1, Floor[l/2] } ] ); Table[ f[n], {n, 0, 100} ]

from sympy import floor, ceiling
def A064834(n):
    x, y = str(n), 0
    lx2 = len(x)/2
    for a,b in zip(x[:floor(lx2)],x[:ceiling(lx2)-1:-1]):
        y += abs(int(a)-int(b))
    return y
# Chai Wah Wu, Aug 09 2014

More terms from Vladeta Jovovic, Matthew Conroy and Robert G. Wilson v, Oct 26 2001

A241173 Start with n; add to it any of its digits; repeat; a(n) = minimal number of steps needed to reach a palindrome.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 2, 3, 2, 1, 4, 3, 2, 1, 1, 0, 3, 3, 2, 3, 3, 2, 4, 1, 3, 3, 0, 2, 2, 2, 1, 3, 2, 1, 2, 1, 3, 0, 2, 2, 3, 4, 2, 1, 4, 3, 2, 2, 0, 4, 3, 1, 3, 1, 4, 3, 1, 2, 2, 0, 3, 4, 3, 1, 3, 2, 2, 4, 2, 3, 0, 3, 1, 1, 3, 2, 3, 1, 2, 2, 3, 0, 5, 1, 2, 1, 4, 5, 2, 3, 4, 5, 0

Offset: 0

N. J. A. Sloane, Apr 23 2014

a(n) = 0 iff n is already a palindrome (A002113).
Is it a theorem that a(n) always exists?
a(n) always exists. Proof: A palindrome can be reached by simply adding the initial digit until a palindrome with the same number of digits as the initial number is reached: If no palindrome is reached by then, this will yield a number with initial digit '1'. Thereafter, this procedure will yield the next larger palindrome - either not larger than 19...91 or, after 19...9 + 1 = 20...0, at 20...02. - M. F. Hasler, Apr 26 2014

			Examples for a(10) through a(23):
a(10) = 1 via 10 -> 11
a(11) = 0 via 11
a(12) = 3 via 12 -> 13 -> 16 -> 22
a(13) = 2 via 13 -> 16 -> 22
a(14) = 3 via 14 -> 15 -> 16 -> 22
a(15) = 2 via 15 -> 16 -> 22
a(16) = 1 via 16 -> 22
a(17) = 4 via 17 -> 18 -> 19 -> 20 -> 22
a(18) = 3 via 18 -> 19 -> 20 -> 22
a(19) = 2 via 19 -> 20 -> 22
a(20) = 1 via 20 -> 22
a(21) = 1 via 21 -> 22
a(22) = 0 via 22
a(23) = 3 via 23 -> 25 -> 30 -> 33

Eric Angelini, Posting to Sequence Fans Mailing List, Apr 20 2014

David A. Corneth, Table of n, a(n) for n = 0..9999
David A. Corneth, PARI program
David A. Corneth, Steps taken from n as described in name to reach a palindrome

Cf. A002113.
A241174 gives the smallest number that takes n steps to reach a palindrome.
Related sequences: A241173, A241174, A241175, A241176, A241177, A241178, A241179, A241180, A241181, A241182, A241183.

A241173[n_] := Module[{c, nx},
   If[n == IntegerReverse[n], Return[0]];
   c = 1; nx = n;
   While[ ! AnyTrue[nx = Flatten[nx + IntegerDigits[nx]], # == IntegerReverse[#] &], c++];
   Return[c]];
Table[A241173[i], {i, 0, 100}] (* Robert Price, Mar 17 2019 *)

a(n,m=0)={ if( m, my(d); for(i=1,#d=vecsort(digits(n),,12), d[i] && if( m>1, a(n+d[i],m-1) /*&& !print1("/*",[n,d[i],m],"* /")*/, is_A002113(n+d[i])) && return(m)), is_A002113(n) || until(a(n,m++),); m)} \\ Memoization should be implemented to improve performance which remains poor esp. for terms just above 10^k+1. - M. F. Hasler, Apr 26 2014

\\ See Corneth link; faster than above. David A. Corneth, Mar 21 2019

More terms from M. F. Hasler, Apr 26 2014

cp's OEIS Frontend

A351717 Numbers whose maximal (or lazy) Lucas representation (A130311) is palindromic.

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A046447 Apart from initial term, composite numbers with the property that the concatenation of their prime factors is a palindrome.

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A083852 Decimal palindromes that are multiples of 11.

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A228730 Lexicographically earliest sequence of distinct nonnegative integers such that the sum of two consecutive terms is a palindrome in base 10.

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A260254 Number of ways to write n as sum of two palindromes in decimal representation.

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Haskell

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A261675 Minimal number of palindromes in base 10 that add to n.

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PARI

A332121 a(n) = 2*(10^(2n+1)-1)/9 - 10^n.

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Maple

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A043040 Numbers that are palindromic and divisible by 5.