A180410 -id:A180410 - cp's OEIS frontend

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

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

A227362 Distinct digits of n arranged in decreasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 21, 31, 41, 51, 61, 71, 81, 91, 20, 21, 2, 32, 42, 52, 62, 72, 82, 92, 30, 31, 32, 3, 43, 53, 63, 73, 83, 93, 40, 41, 42, 43, 4, 54, 64, 74, 84, 94, 50, 51, 52, 53, 54, 5, 65, 75, 85, 95, 60, 61, 62, 63, 64, 65, 6, 76, 86

Offset: 0

Reinhard Zumkeller, Jul 09 2013

a(n) <= 9876543210; a(a(n)) = a(n);
A055642(a(n)) <= 10;
A055642(a(n)) <= A055642(n), A055642(a(n)) = A055642(n) iff A178788(n) = 1;
a(A109303(n)) < A109303(n); a(A009995(n)) = A009995(n); a(A071589(n)) > A071589(n);
a(n) = A151949(n) + A180410(n).

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

Cf. A004186, A230959.

import Data.List (nub, sort)
a227362 = read . reverse . sort . nub . show :: Integer -> Integer

a:= n-> parse(cat(sort([{convert(n, base, 10)[]}[]], `>`)[])):
seq(a(n), n=0..68);  # Alois P. Heinz, Sep 21 2022

f[n_] := FromDigits[Reverse@ Union@ IntegerDigits@ n]; f /@ Range[0, 68] (* Michael De Vlieger, Apr 16 2015, corrected by Robert G. Wilson v *)

a(n) = {if (n == 0, d = [0], d = digits(n)); eval(subst(Pol(vecsort(d,,12)), x, 10));} \\ Michel Marcus, Apr 16 2015

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

def A227362(n): return int(''.join(sorted(set(str(n)),reverse=True))) # Chai Wah Wu, Nov 23 2022

A217928 Sum of distinct decimal digits appearing in n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 2, 5, 6, 7, 8, 9, 10, 11, 3, 4, 5, 3, 7, 8, 9, 10, 11, 12, 4, 5, 6, 7, 4, 9, 10, 11, 12, 13, 5, 6, 7, 8, 9, 5, 11, 12, 13, 14, 6, 7, 8, 9, 10, 11, 6, 13, 14, 15, 7, 8, 9, 10, 11, 12, 13, 7, 15, 16, 8, 9, 10, 11, 12, 13, 14, 15, 8, 17, 9, 10, 11, 12, 13, 14, 15, 16, 17, 9, 1

Offset: 0

Max Alekseyev, Oct 15 2012

a(n) <= 45 = 1+2+3+4+5+6+7+8+9; A227378(n) = smallest number m such that a(m) = n, 0 <= n <= 45. - Reinhard Zumkeller, Jul 09 2013

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for Colombian or self numbers and related sequences

Cf. A007953, A180410.

import Data.List (nub)
a217928 = sum . nub . map (read . return) . show :: Integer -> Integer
-- Reinhard Zumkeller, Jul 09 2013

a:= n-> add(i, i={convert(n, base, 10)[]}):
seq(a(n), n=0..100);  # Alois P. Heinz, Sep 21 2022

{ a(n) = local( d = vecsort( eval(Vec(Str(n))),,8) ); sum(i=1,#d,d[i]) }

a(n) = vecsum(Set(digits(n))); \\ Michel Marcus, Sep 13 2022

def a(n): return sum(map(int, set(str(n))))
print([a(n) for n in range(101)]) # Michael S. Branicky, Sep 13 2022

a(n) = A007953(A180410(n)). - Michel Marcus, Sep 21 2022

A237851 a(1)=1; a(n) is the smallest integer not yet in the sequence divisible by all nonzero digits of a(n-1).

Original entry on oeis.org

1, 2, 4, 8, 16, 6, 12, 10, 3, 9, 18, 24, 20, 14, 28, 32, 30, 15, 5, 25, 40, 36, 42, 44, 48, 56, 60, 54, 80, 64, 72, 70, 7, 21, 22, 26, 66, 78, 112, 34, 84, 88, 96, 90, 27, 98, 144, 52, 50, 35, 45, 100, 11, 13, 33, 39, 63, 102, 38, 120, 46, 108, 104, 68, 168

Offset: 1

Eric Angelini, Feb 14 2014

A permutation of the naturals with inverse A237860.

If a(n) is a prime greater than 7 then no digit of a(n-1) is greater than 1, cf. A007088.

Lars Blomberg and Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A002796, A180410, A002473, A237860, A007088.

import Data.List (nub, sort, delete)
a237851 n = a237851_list !! (n-1)
a237851_list = 1 : f 1 [2..] where
   f x zs = g zs where
     g (u:us) | all ((== 0) . (mod u)) ds = u : f u (delete u zs)
              | otherwise = g us
              where ds = dropWhile (<= 1) $
                         sort $ nub $ map (read . return) $ show x
-- Reinhard Zumkeller, Feb 14 2014

a[1] = 1;
a[n_] := a[n] = For[k = 1, True, k++, If[FreeQ[Array[a, n-1], k], If[ AllTrue[Select[IntegerDigits[a[n-1]], #>0&] // Union, Divisible[k, #]&], Return[k]]]];
a /@ Range[100] (* Jean-François Alcover, Nov 26 2019 *)

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

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A227362 Distinct digits of n arranged in decreasing order.

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A217928 Sum of distinct decimal digits appearing in n.

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Haskell

Maple

PARI

PARI

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A237851 a(1)=1; a(n) is the smallest integer not yet in the sequence divisible by all nonzero digits of a(n-1).

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