cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A193460 Numbers m such that A193459(m) = A000005(m).

Original entry on oeis.org

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, 111, 200, 222, 300, 333, 400, 444, 500, 555, 600, 666, 700, 777, 800, 888, 900, 999, 1000, 1111, 2000, 2222, 3000, 3333, 4000, 4444, 5000, 5555, 6000
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 26 2011

Keywords

Comments

Union of A037124 and A010785, apart from initial 0.

Programs

  • Haskell
    import Data.List (elemIndices)
a193460 n = a193460_list !! (n-1)
a193460_list = elemIndices 0 $ 1 : zipWith (-) a193459_list a000005_list

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

References

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

Links

Crossrefs

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

Programs

  • Haskell
    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
  • Magma
    [(n-9*Floor((n-1)/9))*(10^Floor((n+8)/9)-1)/9: n in [0..50]]; // Vincenzo Librandi, Nov 10 2014
  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    a(n)=10^((n+8)\9)\9*((n-1)%9+1) \\ Charles R Greathouse IV, Jun 15 2011
  • PARI
    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
  • PARI
    is(n)={1==#Set(digits(n))}
inv(n) = 9*#Str(n) + n%10 - 9 \\ David A. Corneth, Jun 24 2016
  • Python
    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
  • Python
    print([0]+[int(d*r) for r in range(1, 7) for d in "123456789"]) # Michael S. Branicky, Dec 29 2021
  • Python
    # 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

Formula

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

Extensions

Name clarified by Jon E. Schoenfield, Nov 10 2023

A037124 Numbers that contain only one nonzero digit.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000, 10000, 20000, 30000, 40000, 50000, 60000, 70000, 80000, 90000, 100000
Offset: 1

Views

Author

Vasiliy Danilov (danilovv(AT)usa.net), Jun 15 1998

Keywords

Comments

Starting with 1: next greater number not containing the highest digit (see also A098395). - Reinhard Zumkeller, Oct 31 2004
A061116 is a subsequence. - Reinhard Zumkeller, Mar 26 2008
Subsequence of A193460. - Reinhard Zumkeller, Jul 26 2011

Links

Crossrefs

Cf. A000005, A000079, A011279, A038754, A051885, A055640, A061116, A133464, A138707, A140730, A140740, A193459, A193460.

Programs

  • Haskell
    a037124 n = a037124_list !! (n-1)
a037124_list = f [1..9] where f (x:xs) = x : f (xs ++ [10*x])
-- Reinhard Zumkeller, May 03 2011
  • Magma
    [((n mod 9)+1) * 10^Floor(n/9): n in [0..50]]; // Vincenzo Librandi, Nov 11 2014
  • Mathematica
    Table[(10^Floor[(n - 1)/9])*(n - 9*Floor[(n - 1)/9]), {n, 1, 50}] (* José de Jesús Camacho Medina, Nov 10 2014 *)
Array[(Mod[#, 9] + 1) * 10^Floor[#/9] &, 50, 0] (* Paolo Xausa, Oct 10 2024 *)
  • PARI
    is(n)=n>0 && n/10^valuation(n,10)<10 \\ Charles R Greathouse IV, Jan 29 2017
  • Python
    def A037124(n):
    a, b = divmod(n-1,9)
    return 10**a*(b+1) # Chai Wah Wu, Oct 16 2024

Formula

a(n) = (((n - 1) mod 9) + 1) * 10^floor((n - 1)/9). E.g., a(40) = ((39 mod 9) + 1) * 10^floor(39/9) = (3 + 1) * 10^4 = 40000. - Carl R. White, Jan 08 2004
a(n) = A051885(n-1) + 1. - Reinhard Zumkeller, Jan 03 2008, Jul 10 2011
A138707(a(n)) = A000005(a(n)). - Reinhard Zumkeller, Mar 26 2008
From Reinhard Zumkeller, May 26 2008: (Start)
a(n+1) = a(n) + a(n - n mod 9).
a(n) = A140740(n+9, 9). (End)
A055640(a(n)) = 1. - Reinhard Zumkeller, May 03 2011
A193459(a(n)) = A000005(a(n)). - Reinhard Zumkeller, Jul 26 2011
Sum_{n>0} 1/a(n)^s = (10^s)*(zeta(s) - zeta(s,10))/(10^s-1), with (s>1). - Enrique Pérez Herrero, Feb 05 2013
a(n) = (10^floor((n - 1)/9))*(n - 9*floor((n - 1)/9)). - José de Jesús Camacho Medina, Nov 10 2014
From Chai Wah Wu, May 28 2016: (Start)
a(n) = 10*a(n-9).
G.f.: x*(9*x^8 + 8*x^7 + 7*x^6 + 6*x^5 + 5*x^4 + 4*x^3 + 3*x^2 + 2*x + 1)/(1 - 10*x^9). (End)
a(n) ≍ 1.2589...^n, where the constant is A011279. (f ≍ g when f << g and g << f, that is, there are absolute constants c,C > 0 such that for all large n, |f(n)| <= c|g(n)| and |g(n)| <= C|f(n)|.) - Charles R Greathouse IV, Mar 11 2021
Sum_{n>=1} 1/a(n) = 7129/2268. - Amiram Eldar, Jan 21 2022
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.