A178788 -id:A178788 - cp's OEIS frontend

A002113 Palindromes in base 10.

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

A055642 Number of digits in the decimal expansion of n.

Original entry on oeis.org

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

Offset: 0

Henry Bottomley, Jun 06 2000

From Hieronymus Fischer, Jun 08 2012: (Start)
For n > 0 the first differences of A117804.
The total number of digits necessary to write down all the numbers 0, 1, 2, ..., n is A117804(n+1). (End)
Here a(0) = 1, but a different common convention is to consider that the expansion of 0 in any base b > 0 has 0 terms and digits. - M. F. Hasler, Dec 07 2018

			Examples:
999: 1 + floor(log_10(999)) = 1 + floor(2.x) = 1 + 2 = 3 or
      ceiling(log_10(999+1)) = ceiling(log_10(1000)) = ceiling(3) = 3;
1000: 1 + floor(log_10(1000)) = 1 + floor(3) = 1 + 3 = 4 or
      ceiling(log_10(1000+1)) = ceiling(log_10(1001)) = ceiling(3.x) = 4;
1001: 1 + floor(log_10(1001)) = 1 + floor(3.x) = 1 + 3 = 4 or
      ceiling(log_10(1001+1)) = ceiling(log_10(1002)) = ceiling(3.x) = 4;

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A043537, A178788, A046034, A019546, A054899, A122840, A055640, A055641, A102669-A102685, A117804, A160093, A160094, A196563, A196564, A000120, A000788, A023416, A059015 (for base 2).
Cf. A262190, A262198.
Cf. A007953: sum of digits.

a055642 :: Integer -> Int
a055642 = length . show  -- Reinhard Zumkeller, Feb 19 2012, Apr 26 2011

[ #Intseq(n): n in [0..105] ];   //  Bruno Berselli, Jun 30 2011
(Common Lisp) (defun A055642 (n) (if (zerop n) 1 (floor (log n 10)))) ; James Spahlinger, Oct 13 2012

A055642 := proc(n)
        max(1,ilog10(n)+1) ;
end proc:  # R. J. Mathar, Nov 30 2011

Join[{1}, Array[ Floor[ Log[10, 10# ]] &, 104]] (* Robert G. Wilson v, Jan 04 2006 *)
Join[{1},Table[IntegerLength[n],{n,104}]]
IntegerLength[Range[0,120]] (* Harvey P. Dale, Jul 02 2016 *)

a(n)=#Str(n) \\ M. F. Hasler, Nov 17 2008

A055642(n)=logint(n+!n,10)+1 \\ Increasingly faster than the above, for larger n. (About twice as fast for n ~ 10^7.) - M. F. Hasler, Dec 07 2018

def a(n): return len(str(n))
print([a(n) for n in range(121)]) # Michael S. Branicky, May 10 2022

def A055642(n): # Faster than len(str(n)) from ~ 50 digits on
    L = math.log10(n or 1)
    if L.is_integer() and 10**int(L)>n: return int(L or 1)
    return int(L)+1  # M. F. Hasler, Apr 08 2024

a(A046760(n)) < A050252(A046760(n)); a(A046759(n)) > A050252(A046759(n)). - Reinhard Zumkeller, Jun 21 2011
a(n) = A196563(n) + A196564(n).
a(n) = 1 + floor(log_10(n)) = 1 + A004216(n) = ceiling(log_10(n+1)) = A004218(n+1), if n >= 1. - Daniel Forgues, Mar 27 2014
a(A046758(n)) = A050252(A046758(n)). - Reinhard Zumkeller, Jun 21 2011
a(n) = A117804(n+1) - A117804(n), n > 0. - Hieronymus Fischer, Jun 08 2012
G.f.: g(x) = 1 + (1/(1-x))*Sum_{j>=0} x^(10^j). - Hieronymus Fischer, Jun 08 2012
a(n) = A262190(n) for n < 100; a(A262198(n)) != A262190(A262198(n)). - Reinhard Zumkeller, Sep 14 2015

A010784 Numbers with distinct decimal digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 102, 103, 104, 105, 106, 107, 108, 109, 120

Offset: 1

N. J. A. Sloane

More than the usual number of terms are displayed in order to show the difference from some closely related sequences.
Also: a(1) = 0; a(n) = Min{x integer | x > a(n-1) and all digits to base 10 are distinct}.
This sequence is finite: a(8877691) = 9876543210 is the last term; a(8877690) = 9876543201. The largest gap between two consecutive terms before a(249999) = 2409653 is 104691, as a(175289) = 1098765, a(175290) = 1203456. - Reinhard Zumkeller, Jun 23 2001
Complement of A109303. - David Wasserman, May 21 2008
For the analogs in other bases b, search for "xenodromes."  A001339(b-1) is the number of base b xenodromes for b >= 2. - Rick L. Shepherd, Feb 16 2013
A073531 gives the number of positive n-digit numbers in this sequence. Note that it does not count 0. - T. D. Noe, Jul 09 2013
Can be seen as irregular table whose n-th row holds the n-digit terms; length of row n is then A073531(n) = 9*9!/(10-n)! except for n = 1 where we have 10 terms, unless 0 is considered to belong to a row 0. - M. F. Hasler, Dec 10 2018

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Digit

Subsequence of A043096.

Cf. A109303, A029740 (odds), A029741 (evens), A029743 (primes), A001339.

a010784 n = a010784_list !! (n-1)
a010784_list = filter ((== 1) . a178788) [1..]
-- Reinhard Zumkeller, Sep 29 2011

Select[Range[0,100], Max[DigitCount[#]] == 1 &] (* Harvey P. Dale, Apr 04 2013 *)

is(n)=my(v=vecsort(digits(n)));v==vecsort(v,,8) \\ Charles R Greathouse IV, Sep 17 2012

select( is(n)=!n||#Set(digits(n))==logint(n,10)+1, [0..120]) \\ M. F. Hasler, Dec 10 2018

apply( A010784_row(n,L=List(if(n>1,[])))={forvec(d=vector(n,i,[0,9]),forperm(d,p,p[1]&&listput(L,fromdigits(Vec(p)))),2);Set(L)}, [1..2]) \\ A010784_row(n) returns all terms with n digits. - M. F. Hasler, Dec 10 2018

A010784_list = [n for n in range(10**6) if len(set(str(n))) == len(str(n))] # Chai Wah Wu, Oct 13 2019

# alternate for generating full sequence
from itertools import permutations
afull = [0] + [int("".join(p)) for d in range(1, 11) for p in permutations("0123456789", d) if p[0] != "0"]
print(afull[:100]) # Michael S. Branicky, Aug 04 2022

def hasDistinctDigits(n: Int): Boolean = {
  val numerStr = n.toString
  val digitSet = numerStr.split("").toSet
  numerStr.length == digitSet.size
}
(0 to 99).filter(hasDistinctDigits) // Alonso del Arte, Jan 09 2020

A178788(a(n)) = 1; A178787(a(n)) = n; A043537(a(n)) = A055642(a(n)). - Reinhard Zumkeller, Jun 30 2010

A107846(a(n)) = 0. - Reinhard Zumkeller, Jul 09 2013

Offset changed to 1 and first comment adjusted by Reinhard Zumkeller, Jun 14 2010

A050278 Pandigital numbers: numbers containing the digits 0-9. Version 1: each digit appears exactly once.

Original entry on oeis.org

1023456789, 1023456798, 1023456879, 1023456897, 1023456978, 1023456987, 1023457689, 1023457698, 1023457869, 1023457896, 1023457968, 1023457986, 1023458679, 1023458697, 1023458769, 1023458796, 1023458967, 1023458976, 1023459678, 1023459687, 1023459768

Offset: 1

Eric W. Weisstein, Dec 11 1999

This is a finite sequence with 9*9! = 3265920 terms: a(9*9!) = 9876543210.
A171102 is the infinite version, where each digit must appear at least once.
More precisely, this is exactly the subset of the first 9*9! terms of A171102. - M. F. Hasler, Jan 05 2020
Subsequence of A134336 and of A178403; A178401(a(n)) = 1. - Reinhard Zumkeller, May 27 2010
Smallest prime factors: A178775(n) = A020639(a(n)). - Reinhard Zumkeller, Jun 11 2010
A178788(a(n)) = 1. - Reinhard Zumkeller, Jun 30 2010
All these numbers are composite because the sum of the digits, 45, is divisible by 9. - T. D. Noe, Nov 09 2011
This is the 10th row of the array T(k,n) = n-th number in which the number of distinct base-10 digits is k. A031969 is the 4th row. A220063 is the 5th row. A220076 is the 6th row. A218019 is the 7th row. A219743 is the 8th row. - Jonathan Vos Post, Dec 05 2012
From Hieronymus Fischer, Feb 13 2013: (Start)
The sum of all terms is 9!*49444444440 = 17942399998387200.
General formula for the sum of all terms of the finite sequence of the corresponding base-p pandigital numbers with p places: sum = ((p^2 - p - 1)*(p^p - 1) + p - 1)*(p-2)!/2.
General formula for the sum of all terms (interpreted as decimal permutational numbers with exactly d+1 different digits from the range 0..d < 10): sum = (d+1)!*((10d - 1)*10^d - d + 1)/18, d > 1.
(End)

Robert G. Wilson v, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Pandigital Number
Chai Wah Wu, Pandigital and penholodigital numbers, arXiv:2403.20304 [math.GM], 2024. See p. 1.

Cf. A171102, A050288, A050289.
Cf. A199630, A199631, A114260, A199632, A199633.
Cf. A031969, A050278, A220063, A220076, A218019, A219743.

Select[ FromDigits@# & /@ Permutations[ Range[0, 9]], # > 10^9 &, 20] (* Robert G. Wilson v, May 30 2010, Jan 17 2012 *)

A050278(n)={ my(b=vector(9,k,1+(n+9!-1)%(k+1)!\k!), t=b[9]-1, d=vector(9,i,i+(i>t)-1)); for(i=1,8, t=10*t+d[b[9-i]]; d=vecextract(d,Str("^"b[9-i]))); t*10+d[1]} \\ M. F. Hasler, Jan 15 2012

is_A050278(n)={ 9<#vecsort(Vecsmall(Str(n)),,8) & n<1e10 } /* assuming that n is a nonnegative integer */ /* M. F. Hasler, Jan 10 2012 */

a(n)=my(d=numtoperm(10,n+9!-1));sum(i=1,#d,(d[i]-1)*10^(#d-i)) \\ David A. Corneth, Jun 01 2014

from itertools import permutations
A050278_list = [int(''.join(d)) for d in permutations('0123456789',10) if d[0] != '0'] # Chai Wah Wu, May 25 2015

A050278 = 9*A171571. - M. F. Hasler, Jan 12 2012

A050278(n) = A171102(n) for n <= 9*9!.

Edited by N. J. A. Sloane, Sep 25 2010 to clarify that this is a finite sequence

A171102 Pandigital numbers: numbers containing the digits 0-9. Version 2: each digit appears at least once.

Original entry on oeis.org

1023456789, 1023456798, 1023456879, 1023456897, 1023456978, 1023456987, 1023457689, 1023457698, 1023457869, 1023457896, 1023457968, 1023457986, 1023458679, 1023458697, 1023458769, 1023458796, 1023458967, 1023458976

Offset: 1

N. J. A. Sloane, Sep 25 2010

This is the infinite version. See A050278 for the finite version.
The first 9*9!=3265920 terms of this sequence are permutations of the digits 0-9 with a(9*9!)=9876543210 (see Version 1, A050278). - Jeremy Gardiner, May 29 2010
Subsequence of A134336 and of A178403; A178401(a(n))>0. - Reinhard Zumkeller, May 27 2010
Smallest prime factors: A178775(n) = A020639(a(n)). - Reinhard Zumkeller, Jun 11 2010
A178788(a(n)) = 1, for n <= 9*9!, else A178788(a(n)) = 0. - Reinhard Zumkeller, Jun 30 2010 [corrected by Hieronymus Fischer, Feb 02 2013]
A230959(a(n)) = 0. - Reinhard Zumkeller, Nov 02 2013
The first term of the sequence absent in A050278 is a(3265921) = 10123456789. Also, the  first prime is a(3306373) = 10123457689 = A050288(1). - Zak Seidov, Sep 23 2015
Almost all numbers are in this sequence, in the sense that it has asymptotic density equal to 1. Indeed, the fraction of n-digit numbers which don't have a given digit d is roughly 0.9^n (not exactly because the first digit is chosen among {1..9}) which tends to zero as n -> oo. - M. F. Hasler, Jan 05 2020

Robert G. Wilson v, Table of n, a(n) for n = 1..1000 . [From _Robert G. Wilson v_, May 30 2010]
Eric Weisstein's World of Mathematics, Pandigital Number.
Chai Wah Wu, Pandigital and penholodigital numbers, arXiv:2403.20304 [math.GM], 2024. See p. 1.

Cf. A050278, A050288, A050289.
Cf. A051264, A051018, A051019, A051020.
Subsequence of A253172.

Take[ Select[ FromDigits@# & /@ Permutations[ Range[0, 9], {10}], # > 10^9 &], 20] (* Robert G. Wilson v, May 30 2010 *)

is_A171102(n)=9<#vecsort(Vecsmall(Str(n)),,8) /* assuming that n is a nonnegative integer. In PARI/GP V.2.4 - 2.9 this is faster than other possibilities involving Set(),Vec(),eval() or digits() */ \\ M. F. Hasler, Jan 10 2012, Sep 19 2017

A171102=A050278 /*** valid for n <= 9*9! ***/ \\ M. F. Hasler, Jan 10 2012

a(n) = 1011111111 + A178478(n) for n = 1,...,8!. - M. F. Hasler, Jan 10 2012

A171102(n) = A050278(n) for n <= 9*9!.

A043537 Number of distinct base-10 digits of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2

Offset: 1

Clark Kimberling

a(A000079(A130694(n))) = 10. - Reinhard Zumkeller, Jul 29 2007
a(A000290(A016070(n))) = 2. - Reinhard Zumkeller, Aug 05 2010
a(n) = 10 for almost all n. - Charles R Greathouse IV, Oct 02 2013

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

Cf. A047726, A118668, A119797, A119798, A119799, A119823, A038378, A076493, A055642, A178788.

import Data.List (nub)
a043537 = length . nub . show  -- Reinhard Zumkeller, Apr 16 2011

A043537 := proc(n)
        convert(convert(n,base,10),set) ;
        nops(%) ;
end proc: # R. J. Mathar, Dec 22 2012

Count[DigitCount@ #, n_ /; n > 0] & /@ Range@ 120 (* Michael De Vlieger, Aug 29 2015 *)

a(n)=#vecsort(digits(n),,8) \\ Charles R Greathouse IV, Oct 02 2013

def A043537(n): return len(set(str(n))) # Dimiter Skordev, Oct 02 2021

A029743 Primes with distinct digits.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 103, 107, 109, 127, 137, 139, 149, 157, 163, 167, 173, 179, 193, 197, 239, 241, 251, 257, 263, 269, 271, 281, 283, 293, 307, 317, 347, 349, 359, 367, 379, 389

Offset: 1

Patrick De Geest

This sequence has 283086 terms, the last being 987654103 = A007810(9). - Jud McCranie

Intersection of A010784 and A000040; A178788(a(n)) * A010051(a(n)) = 1. [Reinhard Zumkeller, Sep 25 2011]

Reinhard Zumkeller, Table of n, a(n) for n = 1..283086, full sequence
Patrick De Geest, World!Of Numbers

Cf. A000040, A007810, A010051, A010784, A178788.

a029743 n = a029743_list !! (n-1)
a029743_list = filter ((== 1) . a010051) a010784_list
-- Reinhard Zumkeller, Sep 25 2011

t={};Do[p=Prime[n];If[Select[Transpose[Tally[IntegerDigits[p]]][[2]],#>1 &]=={},AppendTo[t,p]],{n,77}];t (* Jayanta Basu, May 04 2013 *)
Select[Prime[Range[80]],Max[DigitCount[#]]<2&] (* Harvey P. Dale, Sep 13 2020 *)

from sympy import isprime
from itertools import permutations as P
dist = [p for d in range(1, 11) for p in P("0123456789", d) if p[0] != "0"]
afull = [t for t in (int("".join(p)) for p in dist) if isprime(t)]
print(afull[:100]) # Michael S. Branicky, Aug 04 2022

A009995 Numbers with digits in strictly decreasing order. From the Macaulay expansion of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 21, 30, 31, 32, 40, 41, 42, 43, 50, 51, 52, 53, 54, 60, 61, 62, 63, 64, 65, 70, 71, 72, 73, 74, 75, 76, 80, 81, 82, 83, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 96, 97, 98, 210, 310, 320, 321, 410, 420, 421, 430, 431, 432, 510, 520, 521, 530

Offset: 1

N. J. A. Sloane

There are precisely 1023 terms (corresponding to every nonempty subset of {0..9}).
A178788(a(n)) = 1. - Reinhard Zumkeller, Jun 30 2010
A193581(a(n)) > 0 for n > 9. - Reinhard Zumkeller, Aug 10 2011
A227362(a(n)) = a(n). - Reinhard Zumkeller, Jul 09 2013
For a fixed natural number r, any natural number n has a unique "Macaulay expansion" n = C(a_r,r)+C(a_{r-1},r-1)+...+C(a_1,1) with a_r > a_{r-1} > ... > a_1 >= 0. If r=10, concatenating the digits a_r, ..., a_1 gives the present sequence. The representation is valid for all n, but the concatenation only makes sense if all the a_i are < 10. - N. J. A. Sloane, Apr 05 2014
a(n) = A262557(A263327(n)); a(A263328(n)) = A262557(n). - Reinhard Zumkeller, Oct 15 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..1023
B. Sury, Macaulay Expansion, Amer. Math. Monthly 121 (2014), no. 4, 359--360. MR3183022. [See p. 359. - _N. J. A. Sloane_, Apr 05 2014]
Eric Weisstein's World of Mathematics, Digit.

Cf. A009993.

Cf. A262557 (sorted lexicographically), A263327, A263328.

import Data.Set (fromList, minView, insert)
a009995 n = a009995_list !! n
a009995_list = 0 : f (fromList [1..9]) where
   f s = case minView s of
         Nothing     -> []
         Just (m,s') -> m : f (foldl (flip insert) s' $
                              map (10*m +) [0..m `mod` 10 - 1])
-- Reinhard Zumkeller, Aug 10 2011

Sort@ Flatten@ Table[FromDigits /@ Subsets[ Range[9, 0, -1], {n}], {n, 10}] (* Zak Seidov, May 10 2006 *)

is(n)=fromdigits(vecsort(digits(n),,12))==n \\ Charles R Greathouse IV, Apr 16 2015

A009993 Numbers whose decimal digits are in strictly increasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 23, 24, 25, 26, 27, 28, 29, 34, 35, 36, 37, 38, 39, 45, 46, 47, 48, 49, 56, 57, 58, 59, 67, 68, 69, 78, 79, 89, 123, 124, 125, 126, 127, 128, 129, 134, 135, 136, 137, 138, 139, 145, 146, 147, 148, 149, 156, 157

Offset: 1

N. J. A. Sloane

Sequence has 512 terms, since every term except 0 corresponds to a nonempty subset of {1,2,3,4,5,6,7,8,9}.
A178788(a(n)) = 1. - Reinhard Zumkeller, Jun 30 2010
Number of terms in [10^(n-1), 10^n): 0, 9, 36, 84, 126, 126, 84, 36, 9, 1. - Robert G. Wilson v, Jul 20 2014

Zak Seidov, Table of n, a(n) for n = 1..512 (full sequence)
Eric Weisstein's World of Mathematics, Digit

Cf. A009995.

Flatten@Table[FromDigits/@Subsets[Range[1, 9], {n}], {n, 0, 9}] (* Zak Seidov, May 19 2006 *)

is_A009993(n)=Set(n=digits(n))==n \\ M. F. Hasler, Dec 11 2019

forsubset(9,s,print1(fromdigits(Vec(s))",")) \\ M. F. Hasler, Dec 11 2019

from itertools import combinations
def afull(): return [0] + sorted(int("".join(c)) for d in range(1, 10) for c in combinations("123456789", d))
print(afull()) # Michael S. Branicky, Sep 16 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

A002113 Palindromes in base 10.

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A055642 Number of digits in the decimal expansion of n.

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A010784 Numbers with distinct decimal digits.

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A050278 Pandigital numbers: numbers containing the digits 0-9. Version 1: each digit appears exactly once.

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