A002113 -id:A002113 - cp's OEIS frontend

A050683 Number of nonzero palindromes of length n.

9, 9, 90, 90, 900, 900, 9000, 9000, 90000, 90000, 900000, 900000, 9000000, 9000000, 90000000, 90000000, 900000000, 900000000, 9000000000, 9000000000, 90000000000, 90000000000, 900000000000, 900000000000, 9000000000000

Offset: 1

Patrick De Geest, Aug 15 1999

In general the number of base k palindromes with n digits is (k-1)*k^floor((n-1)/2). (See A117855 or A225367 for an explanation.) - Henry Bottomley, Aug 14 2000

This sequence does not count 0 as palindrome with 1 digit, see A070252 = (10,9,90,90,...) for the variant which does. - M. F. Hasler, Nov 16 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Dr. Math, More info 1.
Dr. Math, More info 2.
Index entries for linear recurrences with constant coefficients, signature (0,10).

Cf. A002113, A050250, A050251, A070252, A070199.
Cf. A016116 for numbers of binary palindromes, A016115 for prime palindromes.
Cf. A117855 for the base 3 version, and A225367 for a variant.

a:=[9,9];; for n in [3..30] do a[n]:=10*a[n-2]; od; a; # Muniru A Asiru, Oct 07 2018

[9*10^Floor((n-1)/2): n in [1..30]]; // Vincenzo Librandi, Aug 16 2011

seq(9*10^floor((n-1)/2),n=1..30); # Muniru A Asiru, Oct 07 2018

With[{c=9*10^Range[0,20]},Riffle[c,c]] (* or *) LinearRecurrence[{0,10},{9,9},40] (* Harvey P. Dale, Dec 15 2013 *)

A050683(n)=9*10^((n-1)\2) \\ M. F. Hasler, Nov 16 2008

\\ using M. F. Hasler's is_A002113(n) from A002113
is_A002113(n)={Vecrev(n=digits(n))==n}
for(n=1,8,j=0;for(k=10^(n-1),10^n-1,if(is_A002113(k),j++));print1(j,", ")) \\ Hugo Pfoertner, Oct 03 2018

is_palindrome(x)={my(d=digits(x));for(k=1,#d\2,if(d[k]!=d[#d+1-k],return(0)));return(1)}
for(n=1,8,j=0;for(k=10^(n-1),10^n-1,if(is_palindrome(k),j++));print1(j,", ")) \\ Hugo Pfoertner, Oct 02 2018

a(n) = if(n<3, 9, 10*a(n-2)); \\ Altug Alkan, Oct 03 2018

def A050683(n): return 9*10**(n-1>>1) # Chai Wah Wu, Jul 30 2025

a(n) = 9*10^floor((n-1)/2).
From Colin Barker, Apr 06 2012: (Start)
a(n) = 10*a(n-2).
G.f.: 9*x*(1+x)/(1-10*x^2). (End)
E.g.f.: 9*(cosh(sqrt(10)*x) + sqrt(10)*sinh(sqrt(10)*x) - 1)/10. - Stefano Spezia, Jun 11 2022

A061917 Either a palindrome or becomes a palindrome if trailing 0's are omitted.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 22, 30, 33, 40, 44, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 101, 110, 111, 121, 131, 141, 151, 161, 171, 181, 191, 200, 202, 212, 220, 222, 232, 242, 252, 262, 272, 282, 292, 300, 303, 313, 323, 330, 333, 343, 353, 363, 373, 383, 393, 400, 404

Offset: 1

N. J. A. Sloane, Jun 27 2001

Numbers that are palindromes when written with a suitable number of leading zeros. - Jeppe Stig Nielsen, Jan 17 2022

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A002113, A057890, A057891.

a061917 n = a061917_list !! (n-1)
a061917_list = filter chi [0..] where
   chi x = zs == reverse zs where
      zs = dropWhile (== '0') $ reverse $ show x
-- Reinhard Zumkeller, Sep 25 2011

PaleQ[n_Integer, base_Integer] := Module[{idn, trim = n/base^IntegerExponent[n, base]}, idn = IntegerDigits[trim, base]; idn == Reverse[idn]]; Select[Range[0, 500], PaleQ[#, 10] &] (* Lei Zhou, Dec 13 2013 *)
Join[{0},Select[Range[500],PalindromeQ[FromDigits[Drop[IntegerDigits[#],-IntegerExponent[#,10]]]]&]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 27 2017 *)

isOK(k)=k==0||fromdigits(Vecrev(digits(k)))==k/10^valuation(k,10) \\ Jeppe Stig Nielsen, Jan 17 2022

def ispal(s): return s == s[::-1]
def ok(n): s = str(n); return ispal(s) or ispal(s.rstrip('0'))
print([k for k in range(405) if ok(k)]) # Michael S. Branicky, Jan 17 2022

A136522(A004151(a(n))) = 1. - Reinhard Zumkeller, Sep 25 2011

Corrected by Ray Chandler, Jun 08 2009

A065001 a(n) = (presumed) number of palindromes in the 'Reverse and Add!' trajectory of n, or -1 if this number is not finite.

Original entry on oeis.org

11, 10, 8, 9, 10, 7, 6, 8, 4, 9, 9, 6, 7, 5, 5, 7, 6, 3, 4, 8, 6, 8, 5, 5, 7, 6, 3, 4, 4, 6, 7, 5, 6, 7, 6, 3, 4, 4, 4, 7, 5, 5, 7, 7, 3, 4, 4, 4, 2, 5, 5, 7, 6, 3, 5, 4, 4, 2, 6, 5, 7, 6, 3, 4, 4, 5, 2, 6, 3, 7, 6, 3, 4, 4, 4, 2, 7, 3, 5, 6, 3, 4, 4, 4, 2, 6, 3, 6, 1, 3, 4, 4, 4, 2, 6, 3, 5, 1, 3, 8, 8, 6, 6

Offset: 1

Klaus Brockhaus, Nov 01 2001

Presumably a(196) = 0 (see A016016). Conjecture: There is no n > 0 such that the trajectory of n contains an infinite number of palindromes; the trajectory of n eventually leads to a term in the trajectory of some integer k which belongs to sequence A063048, i.e. whose trajectory (presumably) never leads to a palindrome.

			8, 77, 1111, 2222, 4444, 8888, 661166, 3654563 are the eight palindromes in the trajectory of 8 and 3654563 + 3654563 = 7309126 is the sixth term in the trajectory of 10577 (see A063433) which (presumably) never leads to a palindrome (see A063048), so a(8) = 8.

Index entries for sequences related to Reverse and Add!

A002113, A016016, A033865, A023108, A063048, A063433.

maxarg := 120; stop := 500; for k := 1 to maxarg do n := k; count := 0; c := 0; while c < stop do if n = int_reverse(n) then inc(count); c := 0; end; inc(c); n := n + int_reverse(n); end; write(count," " ); end;

A077594 Smallest number whose Reverse and Add! trajectory (presumably) contains exactly n palindromes, or -1 if there is no such number.

Original entry on oeis.org

196, 89, 49, 18, 9, 14, 7, 6, 3, 4, 2, 1, 10000, -1, -1, -1, -1, -1, -1, -1, -1

Offset: 0

Klaus Brockhaus, Nov 08 2002

Conjecture 1: For each k > 0 the trajectory of k eventually leads to a term in the trajectory of some j which belongs to A063048, i.e. whose trajectory (presumably) never leads to a palindrome. Conjecture 2: There is no k > 0 such that the trajectory of k contains more than twelve palindromes, i.e. a(n) = -1 for n > 12.

			a(9) = 4 since the trajectory of 4 contains the nine palindromes 4, 8, 77, 1111, 2222, 4444, 8888, 661166, 3654563 and at 7309126 joins the trajectory of 10577 = A063048(6) and no m < 4 contains exactly nine palindromes.

Index entries for sequences related to Reverse and Add!

Cf. A002113, A006960, A023108, A063048, A063433, A065001, A070742.

A115683 Numbers that are the product of 2 palindromes greater than 1.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32, 33, 35, 36, 40, 42, 44, 45, 48, 49, 54, 55, 56, 63, 64, 66, 72, 77, 81, 88, 99, 110, 121, 132, 154, 165, 176, 198, 202, 220, 222, 231, 242, 262, 264, 275, 282, 297, 302, 303, 308, 322, 330, 333

Offset: 1

Giovanni Resta, Jan 31 2006

			262 = 2 * 131.
264 = 6 * 11.
275 = 5 * 55.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002113, A141322.

revdigs:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
N:=4; # to get terms <= 2*10^N
Pals:= $2..9:
for d from 2 to N do
  if d::even then
    m:= d/2;
    Pals:= Pals, seq(n*10^m + revdigs(n), n=10^(m-1)..10^m-1);
  else
    m:= (d-1)/2;
    Pals:= Pals, seq(seq(n*10^(m+1)+y*10^m+revdigs(n), y=0..9), n=10^(m-1)..10^m-1);
  fi
od:
Pals:= {Pals}: nP:= nops(Pals):
P2:= select(`<`,{seq(seq(Pals[i]*Pals[j],j=1..i),i=1..nP)},2*10^N):
sort(convert(P2,list)); # Robert Israel, Mar 16 2020

pal = Select[ Range[2, 400], # == FromDigits@ Reverse@ IntegerDigits@ # &]; Select[Union[Times @@@ Tuples[pal, 2]], # <= 400 &] (* Giovanni Resta, Jun 20 2016 *)

A259380 Palindromic numbers in bases 2 and 8 written in base 10.

Original entry on oeis.org

0, 1, 3, 5, 7, 9, 27, 45, 63, 65, 73, 195, 219, 325, 341, 365, 381, 455, 471, 495, 511, 513, 585, 1539, 1755, 2565, 2709, 2925, 3069, 3591, 3735, 3951, 4095, 4097, 4161, 4617, 4681, 12291, 12483, 13851, 14043, 20485, 20613, 20805, 20933, 21525, 21653, 21845, 21973, 23085, 23213, 23405, 23533, 24125, 24253, 24445, 24573, 28679, 28807, 28999, 29127, 29719, 29847

Offset: 1

Eric A. Schmidt and Robert G. Wilson v, Jul 16 2015

			2709 is in the sequence because 2709_10 = 5225_8 = 101010010101_2.

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

Cf. A048268, A060792, A097856, A097928, A182232, A259374, A097929, A182233, A259375, A259376,
A097930, A182234, A259377, A259378, A249156, A097931, A259380, A259381, A259382, A259383,
A259384, A099145, A259385, A259386, A259387, A259388, A259389, A259390, A099146, A007632,
A007633, A029961, A029962, A029963, A029964, A029804, A029965, A029966, A029967, A029968,
A029969, A029970, A029731, A097855, A250408, A250409, A250410, A250411, A099165, A250412.

(* first load nthPalindromeBase from A002113 *) palQ[n_Integer, base_Integer] := Block[{}, Reverse[ idn = IntegerDigits[n, base]] == idn]; k = 0; lst = {}; While[k < 21000000, pp = nthPalindromeBase[k, 8]; If[palQ[pp, 2], AppendTo[lst, pp]; Print[pp]]; k++]; lst
b1=2; b2=8; lst={}; Do[d1=IntegerDigits[n, b1]; d2=IntegerDigits[n, b2]; If[d1==Reverse[d1]&&d2==Reverse[d2], AppendTo[lst, n]], {n, 0, 30000}]; lst (* Vincenzo Librandi, Jul 17 2015 *)

Intersection of A006995 and A029803.

A265640 Prime factorization palindromes (see comments for definition).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 25, 27, 28, 29, 31, 32, 36, 37, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 89, 92, 97, 98, 99, 100, 101, 103, 107, 108, 109, 112, 113, 116, 117, 121, 124, 125, 127, 128, 131, 137, 139, 144

Offset: 1

Vladimir Shevelev, Dec 11 2015

nonn

a(66) is the first term at which this sequence differs from A119848.
A number N is called a prime factorization palindrome (PFP) if all its prime factors, taking into account their multiplicities, can be arranged in a row with central symmetry (see example). It is easy to see that every PFP-number is either a square or a product of a square and a prime. In particular, the sequence contains all primes.
Numbers which are both palindromes (A002113) and PFP are 1,2,3,4,5,7,9,11,44,99,101,... (see A265641).
If n is in the sequence, so is n^k for all k >= 0. - Altug Alkan, Dec 11 2015
The sequence contains all perfect numbers except 6 (cf. A000396). - Don Reble, Dec 12 2015
Equivalently, numbers that have at most one prime factor with odd multiplicity. - Robert Israel, Feb 03 2016
Numbers whose squarefree part is noncomposite. - Peter Munn, Jul 01 2020

			44 is a member, since 44=2*11*2.
52 is a member, since 52=2*13*2. [This illustrates the fact that the digits don't need to form a palindrome. This is not a base-dependent sequence. - _N. J. A. Sloane_, Oct 05 2024]
180 is a member, since 180=2*3*5*3*2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000396, A000720, A002113, A265641, complement of A229153.
Disjoint union of A229125 and (A000290\{0}).
Cf. A013661 (zeta(2)).

N:= 1000: # to get all terms <= N
P:= [1,op(select(isprime, [2,seq(i,i=3..N,2)]))]:
sort([seq(seq(p*x^2,x=1..floor(sqrt(N/p))),p=P)]); # Robert Israel, Feb 03 2016

M = 200; P = Join[{1}, Select[Join[{2}, Range[3, M, 2]], PrimeQ]]; Sort[ Flatten[Table[Table[p x^2, {x, 1, Floor[Sqrt[M/p]]}], {p, P}]]] (* Jean-François Alcover, Apr 09 2019, after Robert Israel *)

for(n=1, 200, if( ispseudoprime(core(n)) || issquare(n), print1(n, ", "))) \\ Altug Alkan, Dec 11 2015

from math import isqrt
from sympy.ntheory.factor_ import core, isprime
def ok(n): return isqrt(n)**2 == n or isprime(core(n))
print([k for k in range(1, 145) if ok(k)]) # Michael S. Branicky, Oct 03 2024

from math import isqrt
from sympy import primepi, mobius
def A265640(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c = n-(a:=isqrt(x))
        for y in range(1,a+1):
            m = x//y**2
            c -= primepi(m)-sum(mobius(k)*(m//k**2) for k in range(1, isqrt(m)+1))
        return c
    return bisection(f,n,n) # Chai Wah Wu, Jan 30 2025

lim A(x)/pi(x) = zeta(2) where A(x) is the number of a(n) <= x and pi is A000720.

A332112 a(n) = (10^(2n+1)-1)/9 + 10^n.

Original entry on oeis.org

2, 121, 11211, 1112111, 111121111, 11111211111, 1111112111111, 111111121111111, 11111111211111111, 1111111112111111111, 111111111121111111111, 11111111111211111111111, 1111111111112111111111111, 111111111111121111111111111, 11111111111111211111111111111, 1111111111111112111111111111111

Offset: 0

M. F. Hasler, Feb 09 2020

a(0) = 2 is the only prime in this sequence, since all other terms factor as a(n) = R(n+1)*(10^n+1), where R(n) = (10^n-1)/9.

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

Cf. A002275 (repunits R_n = (10^n-1)/9), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332132 .. A332192 (variants with different repeated digit 3, ..., 9).
Cf. A332113 .. A332119 (variants with different middle digit 3, ..., 9).
Cf. A331860 & A331861 (indices of primes in non-palindromic variants).

Maple
```
A332112 := n -> (10^(2*n+1)-1)/9+10^n;
```

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

apply( {A332112(n)=10^(n*2+1)\9*1+10^n}, [0..15])

def A332112(n): return 10**(n*2+1)//9+10**n

a(n) = A138148(n) + 2*10^n = A002275(2n+1) + 10^n.
G.f.: (2 - 101*x)/((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.

A046328 Palindromes with exactly 2 prime factors (counted with multiplicity).

Original entry on oeis.org

4, 6, 9, 22, 33, 55, 77, 111, 121, 141, 161, 202, 262, 303, 323, 393, 454, 505, 515, 535, 545, 565, 626, 707, 717, 737, 767, 818, 838, 878, 898, 939, 949, 959, 979, 989, 1111, 1441, 1661, 1991, 3113, 3223, 3443, 3883, 7117, 7447, 7997, 9119, 9229, 9449, 10001

Offset: 1

Patrick De Geest, Jun 15 1998

			111 is a palindrome and 111 = 3*37. 3 and 37 are primes.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 2000 terms from Zak Seidov, terms a(2001)-a(2816) from Michael De Vlieger)

Cf. A046315, A046408, A108505.

Subsequence of A001358 and A046338.

fQ[n_] := Block[{id = IntegerDigits[n]}, Plus @@ Last /@ FactorInteger[n] == 2 && id == Reverse[id]]; Select[ Range[ 10000], fQ[ # ] &] (* Robert G. Wilson v, Jun 06 2005 *)
Select[Range[10002], Reverse[x = IntegerDigits[#]] == x && PrimeOmega[#] == 2 &] (* Jayanta Basu, Jun 23 2013 *)
Select[Range[11000],PalindromeQ[#]&&PrimeOmega[#]==2&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 30 2018 *)

ispal(n) = my(d=digits(n));d == Vecrev(d) \\ A002113
for(k=1,1e4,if(ispal(k)&&bigomega(k)==2, print1(k, ", "))) \\ Alexandru Petrescu, Jul 07 2022

from sympy import factorint
from itertools import product
def ispal(n): s = str(n); return s == s[::-1]
def pals(d, base=10): # all d-digit palindromes
    digits = "".join(str(i) for i in range(base))
    for p in product(digits, repeat=d//2):
        if d > 1 and p[0] == "0": continue
        left = "".join(p); right = left[::-1]
        for mid in [[""], digits][d%2]: yield int(left + mid + right)
def ok(pal): return sum(factorint(pal).values()) == 2
print(list(filter(ok, (p for d in range(1, 6) for p in pals(d) if ok(p))))) # Michael S. Branicky, Aug 14 2022

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).

cp's OEIS Frontend

A050683 Number of nonzero palindromes of length n.

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A061917 Either a palindrome or becomes a palindrome if trailing 0's are omitted.

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A065001 a(n) = (presumed) number of palindromes in the 'Reverse and Add!' trajectory of n, or -1 if this number is not finite.

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ARIBAS

A077594 Smallest number whose Reverse and Add! trajectory (presumably) contains exactly n palindromes, or -1 if there is no such number.

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A115683 Numbers that are the product of 2 palindromes greater than 1.

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Maple

Mathematica

A259380 Palindromic numbers in bases 2 and 8 written in base 10.

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A265640 Prime factorization palindromes (see comments for definition).

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