A047842 -id:A047842 - cp's OEIS frontend

A047841 Autobiographical numbers: Fixed under operator T (A047842): "Say what you see".

22, 10213223, 10311233, 10313314, 10313315, 10313316, 10313317, 10313318, 10313319, 21322314, 21322315, 21322316, 21322317, 21322318, 21322319, 31123314, 31123315, 31123316, 31123317, 31123318, 31123319

Offset: 1

Ulrich Schimke (ulrschimke(AT)aol.com)

A digit count numerically summarizes the frequency of digits 0 through 9 in that order when they occur in a number.
This uses a different method from A108810. Here the digits are described in increasing order, whereas in A108810 they can be described in any order.
This sequence is finite, since T(x) < x for every x with at least 22 digits. Last term is a(109) = 101112213141516171819. - Schimke
A character in the Verghese (2009) novel declares that 10213223 "is the only number that describes itself when you read it." - Alonso del Arte, Jan 26 2014

			10313314 contains 1 0's, 3 1's, 3 3's and 1 4's, hence T(10313314) = 10313314 is in the sequence
The entry 3122331418, for instance, is a member since it is indeed made up of three 1's, two 2's, three 3's, one 4 and one 8.

J. N. Kapur, Reflections of a Mathematician, Chapter 33, pp. 314-318, Arya Book Depot, New Delhi 1996.
Abraham Verghese, Cutting for Stone: A Novel. New York: Alfred A. Knopf (2009): 294.

David Wasserman, Table of n, a(n) for n = 1..109 (full sequence)
Andre Kowacs, Studies on the Pea Pattern Sequence, arXiv:1708.06452 [math.HO], 2017.

Cf. A005151, which is the sequence 1, T(1), T(T(1)), .. ending in the fixed-point 21322314.

Cf. A104785, A104786, A104787, A108810.

Entry revised by N. J. A. Sloane, Dec 15 2006

A237605 Numbers n such that A047842(n) | n.

Original entry on oeis.org

0, 22, 777, 4444, 303300, 333333, 555555, 588588, 666666, 888688, 2032230, 5055555, 5858558, 6568588, 6868288, 7339393, 8282088, 10213223, 10311233, 10313314, 10313315, 10313316, 10313317, 10313318, 10313319, 20002200, 21322314, 21322315, 21322316, 21322317, 21322318

Offset: 1

Paolo P. Lava, Feb 10 2014

Autobiographical numbers (A047841) are a subset of this sequence.

The first 3 terms which contain more than 9 copies of a digit are 666666666666, 898888888898 and 4444044404444. - Giovanni Resta, Feb 10 2014

Giovanni Resta, Table of n, a(n) for n = 1..1000 (first 101 terms from Paolo P. Lava)

Cf. A047841, A047842.

P:=proc(q) local a,b,c,d,f,n,v; print(0); v:=array[0..9];
for n from 1 to q do a:=n; for b from 0 to 9 do v[b]:=0; od;
while a>0 do b:=a mod 10; v[b]:=v[b]+1; a:=trunc(a/10); od; a:=0;
for b from 0 to 9 do if v[b]>0 then c:=10*v[b]+b; f:=0; d:=c;
while d>0 do f:=f+1; d:=trunc(d/10); od; a:=a*10^f+c; fi; od;
if type(n/a,integer) then print(n); fi; od; end: P(10^10);

Select[Range[10^6], Mod[#, FromDigits@ Flatten[IntegerDigits /@ Flatten[ Reverse /@ Tally@ Sort@ IntegerDigits@#]]] == 0 &] (* Giovanni Resta, Feb 10 2014 *)

A278439 Numbers k such that k | A047842(k).

Original entry on oeis.org

1, 2, 5, 22, 105, 188, 258, 327, 663, 15425, 16654, 27848, 40324, 80328, 481263, 10213223, 10311233, 10313314, 10313315, 10313316, 10313317, 10313318, 10313319, 21322314, 21322315, 21322316, 21322317, 21322318, 21322319, 31123314, 31123315, 31123316, 31123317

Offset: 1

Paolo P. Lava, Nov 22 2016

The sequence is bounded. Let us consider a k-digit number n in which all 10 numerals from 0 to 9 are equally distributed: there are k/10 0's, k/10 1's, etc. This is the best case in order to have a number with the greatest number of digits under the transform n -> A047842(n). The number of digits we get is 10 + 10*floor(log_10(k/10) + 1), which must be >= k. The inequality becomes log_10(k/10) >= k/10 - 2, which is solved by k <= 23.75... This means that no term of the sequence can have more than 23 digits.

			A237605(258) = 121518 and 121518/258 = 471.

Paolo P. Lava, List of terms and corresponding ratios for n <= 10^9.

Cf. A047842, A237605, A278440, A278441.

with(numtheory): P:=proc(q) local a,b,c,d,j,k,n; for n from 1 to q do
a:=sort(convert(n,base,10)); k:=1; b:=a[1]; c:=0; for j from 2 to nops(a) do
if a[j]=b then k:=k+1; else d:=10*k+b; c:=c*10^(ilog10(d)+1)+d; k:=1; b:=a[j]; fi; od;
d:=10*k+b; c:=c*10^(ilog10(d)+1)+d; if type(c/n,integer) then print(n); fi; od; end: P(10^10);

a(32) corrected by Sean A. Irvine, May 27 2025

A097601 Differences between A045918 and A047842.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 99, 0, 0, 0, 0, 0, 0, 0, 0, 0, 198, 99, 0, 0, 0, 0, 0, 0, 0, 0, 297, 198, 99, 0, 0, 0, 0, 0, 0, 0, 396, 297, 198, 99, 0, 0, 0, 0, 0, 0, 495, 396, 297, 198, 99, 0, 0, 0, 0, 0, 594, 495, 396, 297, 198, 99, 0, 0, 0, 0, 693, 594, 495, 396, 297, 198, 99, 0, 0

Offset: 0

Odimar Fabeny, Aug 29 2004

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A045918, A047842.

LookAndSayA[n_]:= FromDigits@Flatten@IntegerDigits@Flatten[Through[ {Length, First}[#]] & /@ Split@IntegerDigits@n]; dc[n_]:= FromDigits@ Flatten@Select[Table[{DigitCount[n, 10, k], k}, {k, 0, 9}], #[[1]] > 0 &]; Table[LookAndSayA[n] - dc[n], {n, 0, 100}]

Terms a(31) onward added by G. C. Greubel, Jan 14 2019

A235775 a(n) = A047842(A047842(n)), say what you see, once repeated.

Original entry on oeis.org

1011, 21, 1112, 1113, 1114, 1115, 1116, 1117, 1118, 1119, 1031, 1112, 3112, 3113, 3114, 3115, 3116, 3117, 3118, 3119, 102112, 3112, 22, 211213, 211214, 211215, 211216, 211217, 211218, 211219, 102113, 3113, 211213, 1213, 211314, 211315, 211316, 211317, 211318

Offset: 0

Reinhard Zumkeller, Jan 15 2014

a(n) does not depend on the order of digits of n, a property inherited from A047842. - M. F. Hasler, Jan 11 2024

			a(10) = A047842(1011) = 1031;
a(11) = A047842(21) = 1112;
a(12) = A047842(1112) = 3112;
a(100) = A047842(2011) = 102112;
a(101) = A047842(1021) = 102112;
a(102) = A047842(101112) = 104112.
For n = 20231231, digits of the date 2023-12-31, a(n) = 10213223 = A047842(n) because this is a fixed point of A047842. Since the order of the digits of n does not matter and there are no leading zeros, this holds also for the numbers resulting from notation dd.mm.yyyy or mm/dd/yyyy. - _M. F. Hasler_, Jan 11 2024

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A047842, A328447.

Haskell
```
a235775 = a047842 . a047842
```

A235775(n) = A047842(A047842(n)) \\ M. F. Hasler, Jan 11 2024

def A235775(n):
    s = str(n)
    s = ''.join(str(s.count(d))+d for d in sorted(set(s)))
    return int(''.join(str(s.count(d))+d for d in sorted(set(s)))) # Chai Wah Wu, Feb 12 2023

From M. F. Hasler, Jan 11 2024: (Start)
a(n) = a(A328447(n)) = a(m) for all n and all m having the same digits as n, considering their respective multiplicity.
a(n) = A047842(n) =: m iff m is a fixed point of A047842. (End)

A097599 Differences between A097598 and A047842.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 99, 198, 297, 396, 495, 594, 693, 792, 0, 0, 0, 99, 198, 297, 396, 495, 594, 693, 0

Offset: 0

Odimar Fabeny, Aug 29 2004

Cf. A045918, A047842.

A010785 Repdigit numbers, or numbers whose digits are all equal.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 1111, 2222, 3333, 4444, 5555, 6666, 7777, 8888, 9999, 11111, 22222, 33333, 44444, 55555, 66666, 77777, 88888, 99999, 111111, 222222, 333333, 444444, 555555, 666666

Offset: 0

N. J. A. Sloane

Complement of A139819. - David Wasserman, May 21 2008
Subsequence of A134336 and of A178403. - Reinhard Zumkeller, May 27 2010
Subsequence of A193460. - Reinhard Zumkeller, Jul 26 2011
Intersection of A009994 and A009996. - David F. Marrs, Sep 29 2018
Beiler (1964) called these numbers "monodigit numbers". The term "repdigit numbers" was used by Trigg (1974). - Amiram Eldar, Jan 21 2022

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, New York, 1964, p. 83.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Eric F. Bravo, Carlos A. Gómez and Florian Luca, Product of Consecutive Tribonacci Numbers With Only One Distinct Digit, J. Int. Seq., Vol. 22 (2019), Article 19.6.3.
Eric Fernando Bravo, On concatenations of Padovan and Perrin numbers, Math. Commun. (2023) Vol 28, 105-119.
Mahadi Ddamulira, Repdigits as sums of three balancing numbers, Mathematica Slovaca, (2019), hal-02405969.
Mahadi Ddamulira, Padovan numbers that are concatenations of two distinct repdigits, arXiv:2003.10705 [math.NT], 2020.
Mahadi Ddamulira, Tribonacci numbers that are concatenations of two repdigits, hal-02547159, Mathematics [math] / Number Theory [math.NT], 2020.
Mahadi Ddamulira, Padovan numbers that are concatenations of two distinct repdigits, Mathematica Slovaca, Vol. 71, No. 2 (2021), pp. 275-284.
Bart Goddard and Jeremy Rouse, Sum of two repdigits a square, arXiv:1607.06681 [math.NT], 2016. Mentions this sequence.
Bir Kafle, Florian Luca and Alain Togbé, Triangular Repblocks, Fibonacci Quart., Vol. 56, No. 4 (2018), pp. 325-328.
Bir Kafle, Florian Luca and Alain Togbé, Pentagonal and heptagonal repdigits, Annales Mathematicae et Informaticae, Vol. 52 (2020), pp. 137-145.
Benedict Vasco Normenyo, Bir Kafle, and Alain Togbé, Repdigits as Sums of Two Fibonacci Numbers and Two Lucas Numbers, Integers, Vol. 19 (2019), Article A55.
Salah Eddine Rihane and Alain Togbé, Repdigits as products of consecutive Padovan or Perrin numbers, Arab. J. Math., Vol. 10 (2021), pp. 469-480.
Charles W. Trigg, Infinite sequences of palindromic triangular numbers, The Fibonacci Quarterly, Vol. 12, No. 2 (1974), pp. 209-212.
Eric Weisstein's World of Mathematics, Repdigit.
Wikipedia, Repdigit.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,-10).

Cf. A000005, A009994, A009996, A010879, A037904, A047842, A059995, A065444, A134336, A139819, A151949, A178401, A178403, A180410, A193459, A193460, A202022, A227362, A244112.

a010785 n = a010785_list !! n
a010785_list = 0 : r [1..9] where
   r (x:xs) = x : r (xs ++ [10*x + x `mod` 10])
-- Reinhard Zumkeller, Jul 26 2011

[(n-9*Floor((n-1)/9))*(10^Floor((n+8)/9)-1)/9: n in [0..50]]; // Vincenzo Librandi, Nov 10 2014

A010785 := proc(n)
    (n-9*floor(((n-1)/9)))*((10^(floor(((n+8)/9)))-1)/9) ;
end proc:
seq(A010785(n), n = 0 .. 100); # Robert Israel, Nov 09 2014

fQ[n_]:=Module[{id=IntegerDigits[n]}, Length[Union[id]]==1]; Select[Range[0,10000], fQ] (* Vladimir Joseph Stephan Orlovsky, Dec 29 2010 *)
Union[FromDigits/@Flatten[Table[PadRight[{},i,n],{n,0,9},{i,6}],1]] (* or *) LinearRecurrence[{0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,-10}, {0,1,2,3,4,5,6,7,8,9,11,22,33,44,55,66,77,88},40] (* Harvey P. Dale, Dec 28 2011 *)
Union@ Flatten@ Table[k (10^n - 1)/9, {k, 0, 9}, {n, 6}] (* Robert G. Wilson v, Oct 09 2014 *)
Table[(n - 9 Floor[(n-1)/9]) (10^Floor[(n+8)/9] - 1)/9, {n, 0, 50}] (* José de Jesús Camacho Medina, Nov 06 2014 *)

a(n)=10^((n+8)\9)\9*((n-1)%9+1) \\ Charles R Greathouse IV, Jun 15 2011

nxt(n,t=n%10)=if(t<9,n*(t+1),n*10+9)\t \\ Yields the term a(k+1) following a given term a(k)=n. M. F. Hasler, Jun 24 2016

is(n)={1==#Set(digits(n))}
inv(n) = 9*#Str(n) + n%10 - 9 \\ David A. Corneth, Jun 24 2016

def a(n): return 0 if n == 0 else int(str((n-1)%9+1)*((n-1)//9+1))
print([a(n) for n in range(55)]) # Michael S. Branicky, Dec 29 2021

print([0]+[int(d*r) for r in range(1, 7) for d in "123456789"]) # Michael S. Branicky, Dec 29 2021

# without string operations
def a(n): return 0 if n == 0 else (10**((n-1)//9+1)-1)//9*((n-1)%9+1)
print([a(n) for n in range(55)]) # Michael S. Branicky, Nov 03 2023

A037904(a(n)) = 0. - Reinhard Zumkeller, Dec 14 2007
A178401(a(n)) > 0. - Reinhard Zumkeller, May 27 2010
From Reinhard Zumkeller, Jul 26 2011: (Start)
For n > 0: A193459(a(n)) = A000005(a(n)).
for n > 10: a(n) mod 10 = floor(a(n)/10) mod 10.
A010879(n) = A010879(A059995(n)). (End)
A202022(a(n)) = 1. - Reinhard Zumkeller, Dec 09 2011
a(0)=0, a(1)=1, a(2)=2, a(3)=3, a(4)=4, a(5)=5, a(6)=6, a(7)=7, a(8)=8, a(9)=9, a(10)=11, a(11)=22, a(12)=33, a(13)=44, a(14)=55, a(15)=66, a(16)=77, a(17)=88, a(n) = 11*a(n-9) - 10*a(n-18). - Harvey P. Dale, Dec 28 2011
A151949(a(n)) = 0; A180410(a(n)) = A227362(a(n)). - Reinhard Zumkeller, Jul 09 2013
a(n) = (n - 9*floor((n-1)/9))*(10^floor((n+8)/9) - 1)/9. - José de Jesús Camacho Medina, Nov 06 2014
G.f.: x*(1+2*x+3*x^2+4*x^3+5*x^4+6*x^5+7*x^6+8*x^7+9*x^8)/((1-x^9)*(1-10*x^9)). - Robert Israel, Nov 09 2014
A047842(a(n)) = A244112(a(n)). - Reinhard Zumkeller, Nov 11 2014
Sum_{n>=1} 1/a(n) = (7129/2520) * A065444 = 3.11446261209177581335... - Amiram Eldar, Jan 21 2022

Name clarified by Jon E. Schoenfield, Nov 10 2023

A005151 Summarize the previous term (digits in increasing order), starting with a(1) = 1.

Original entry on oeis.org

1, 11, 21, 1112, 3112, 211213, 312213, 212223, 114213, 31121314, 41122314, 31221324, 21322314, 21322314, 21322314, 21322314, 21322314, 21322314, 21322314, 21322314, 21322314, 21322314, 21322314, 21322314, 21322314, 21322314, 21322314, 21322314, 21322314

Offset: 1

N. J. A. Sloane

a(n) = 21322314 for n > 12. - Reinhard Zumkeller, Jan 25 2014

The digits of each term a(n) are a permutation of those of the corresponding term A063850(n). - Chayim Lowen, Jul 16 2015

			The term after 312213 is obtained by saying "Two 1's, two 2's, two 3's", which gives 21-22-23, i.e., 212223.

C. Fleenor, "A litteral sequence", Solution to Problem 2562, Journal of Recreational Mathematics, vol. 31 No. 4 pp. 307 2002-3 Baywood NY.
Problem in J. Recreational Math., 30 (4) (1999-2000), p. 309.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Colin Barker, Table of n, a(n) for n = 1..1000
V. Bronstein and A. S. Fraenkel, On a curious property of counting sequences, Amer. Math. Monthly, 101 (1994), 560-563.
Onno M. Cain and Sela T. Enin, Inventory Loops (i.e. Counting Sequences) have Pre-period 2 max S_1 + 60, arXiv:2004.00209 [math.NT], 2020.
X. Gourdon and B. Salvy, Effective asymptotics of linear recurrences with rational coefficients, Discrete Mathematics, vol. 153, no. 1-3, 1996, pages 145-163.
James Henle, Is (some) mathematics poetry?, Journal of Humanistic Mathematics 1:1 (2011), pp. 94-100.
Madras Math's Amazing Number Facts, Fact No. 13
Madras Math, Descriptive Number
Trevor Scheopner, The Cyclic Nature (and Other Intriguing Properties) of Descriptive Numbers, Princeton Undergraduate Mathematics Journal, Issue 1, Article 4.
L. J. Upton, Letter to N. J. A. Sloane, Jan 8 1991.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A005150, A047842. See A083671 for another version.

Cf. A023989, A118628, A060857.

import Data.List (group, sort, transpose)
a005151 n = a005151_list !! (n-1)
a005151_list = 1 : f [1] :: [Integer] where
   f xs = (read $ concatMap show ys) : f ys where
          ys = concat $ transpose [map length zss, map head zss]
          zss = group $ sort xs
-- Reinhard Zumkeller, Jan 25 2014

RunLengthEncode[x_List] := (Through[{Length, First}[ #1]] &) /@ Split[ Sort[x]]; LookAndSay[n_, d_:1] := NestList[ Flatten[ RunLengthEncode[ # ]] &, {d}, n - 1]; F[n_] := LookAndSay[n, 1][[n]]; Table[ FromDigits[ F[n]], {n, 25}] (* Robert G. Wilson v, Jan 22 2004 *)
a[1] = 1; a[n_] := a[n] = FromDigits[Reverse /@ Sort[Tally[a[n-1] // IntegerDigits], #1[[1]] < #2[[1]]&] // Flatten]; Array[a, 26] (* Jean-François Alcover, Jan 25 2016 *)

say(n) = {digs = digits(n); d = vecsort(digs,,8); s = ""; for (k=1, #d, nbk = #select(x->x==d[k], digs); s = concat(s, Str(nbk)); s = concat(s, d[k]);); eval(s);}
lista(nn) = {print1(n = 1, ", "); for (k=1, nn, m = say(n); print1(m, ", "); n = m;);} \\ Michel Marcus, Feb 12 2016

a(n,show_all=1,a=1)={for(i=2,n,show_all&&print1(a",");a=A047842(a));a} \\ M. F. Hasler, Feb 25 2018

Vec(x*(1 + 10*x + 10*x^2 + 1091*x^3 + 2000*x^4 + 208101*x^5 + 101000*x^6 - 99990*x^7 - 98010*x^8 + 31007101*x^9 + 10001000*x^10 - 9900990*x^11 - 9899010*x^12) / (1 - x) + O(x^40)) \\ Colin Barker, Aug 23 2018

from itertools import accumulate, groupby, repeat
def summarize(n, _):
  return int("".join(str(len(list(g)))+k for k, g in groupby(sorted(str(n)))))
def aupton(nn): return list(accumulate(repeat(1, nn+1), summarize))
print(aupton(25)) # Michael S. Branicky, Jan 11 2021

a(n+1) = A047842(a(n)). - M. F. Hasler, Feb 25 2018

G.f.: x*(1 + 10*x + 10*x^2 + 1091*x^3 + 2000*x^4 + 208101*x^5 + 101000*x^6 - 99990*x^7 - 98010*x^8 + 31007101*x^9 + 10001000*x^10 - 9900990*x^11 - 9899010*x^12) / (1 - x). - Colin Barker, Aug 23 2018

A244112 Reverse digit count of n in decimal representation.

Original entry on oeis.org

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1110, 21, 1211, 1311, 1411, 1511, 1611, 1711, 1811, 1911, 1210, 1211, 22, 1312, 1412, 1512, 1612, 1712, 1812, 1912, 1310, 1311, 1312, 23, 1413, 1513, 1613, 1713, 1813, 1913, 1410, 1411, 1412, 1413, 24, 1514, 1614, 1714

Offset: 0

Reinhard Zumkeller, Nov 11 2014

Frequencies of digits 0 through 9, occurring in n, are summarized in order of decreasing digits;

a(A010785(n)) = A047842(A010785(n)).

			101 contains two 1s and one 0, therefore a(101) = 2110;
102 contains one 2, one 1 and one 0, therefore a(102) = 121110.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A047842, A010785.

See A036058 for the orbit of 0 under this map.

import Data.List (sort, group); import Data.Function (on)
a244112 :: Integer -> Integer
a244112 n = read $ concat $
   zipWith ((++) `on` show) (map length xs) (map head xs)
   where xs = group $ reverse $ sort $ map (read . return) $ show n

f[n_] := Block[{s = Split@ IntegerDigits@ n}, FromDigits@ Reverse@ Riffle[Union@ Flatten@ s, Length@# & /@ s]]; Array[f, 48, 0] (* Robert G. Wilson v, Dec 01 2016 *)

A244112(n,c=1,S="")={for(i=2,#n=vecsort(digits(n),,4),n[i]==n[i-1]&&c++&&next;S=Str(S,c,n[i-1]);c=1);eval(Str(S,c,if(n,n[#n])))} \\ M. F. Hasler, Feb 25 2018

def A244112(n):
    return int(''.join([str(str(n).count(d))+d for d in '9876543210' if str(n).count(d) > 0])) # Chai Wah Wu, Dec 01 2016

cp's OEIS Frontend

A047841 Autobiographical numbers: Fixed under operator T (A047842): "Say what you see".

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A237605 Numbers n such that A047842(n) | n.

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A278439 Numbers k such that k | A047842(k).

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A097601 Differences between A045918 and A047842.

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A235775 a(n) = A047842(A047842(n)), say what you see, once repeated.

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A097599 Differences between A097598 and A047842.

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A010785 Repdigit numbers, or numbers whose digits are all equal.

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