A078241 -id:A078241 - cp's OEIS frontend

A004290 Least positive multiple of n that when written in base 10 uses only 0's and 1's.

1, 10, 111, 100, 10, 1110, 1001, 1000, 111111111, 10, 11, 11100, 1001, 10010, 1110, 10000, 11101, 1111111110, 11001, 100, 10101, 110, 110101, 111000, 100, 10010, 1101111111, 100100, 1101101, 1110, 111011, 100000, 111111, 111010

Offset: 1

David W. Wilson

It is easy to show that a(n) always exists and in fact has at most n digits [Wu, 2014]. - N. J. A. Sloane, Jun 13 2014
a(n) = min{A007088(k): k > 0 and A007088(k) mod n = 0}. - Reinhard Zumkeller, Jan 10 2012
a(10^k) = 10^k and a(10^k - 1) = (10^(9k) - 1) / 9 for all k. Is a(n) < a(10^k - 1) for all n < 10^k - 1? - David Radcliffe, Aug 01 2025

Chai Wah Wu, Table of n, a(n) for n = 1..9998 (first 2000 terms from T. D. Noe [and Ed Pegg Link])
Ed Pegg Jr., 'Binary' Puzzle.
Eric M. Schmidt, Sage code to compute this sequence.
Chai Wah Wu, Pigeonholes and repunits, Amer. Math. Monthly, 121 (2014), 529-533.

Cf. A004283-A004289, A078241-A078248, A079339, A096681-A096688, A257345.

a004290 0 = 0
a004290 n = head [x | x <- tail a007088_list, mod x n == 0]
-- Reinhard Zumkeller, Jan 10 2012

f:= proc(n)
local L,x,m,r,k,j;
if n<2 then return n fi;
for x from 2 to n-1 do L[0,x]:= 0 od:
L[0,0]:= 1: L[0,1]:= 1;
for m from 1 do
   if L[m-1,(-10^m) mod n] = 1 then break fi;
   L[m,0]:= 1;
   for k from 1 to n-1 do
     L[m,k]:= max(L[m-1,k],L[m-1,k-10^m mod n])
   od;
od;
r:= 10^m; k:= -10^m mod n;
for j from m-1 by -1 to 1 do
    if L[j-1,k] = 0 then
      r:= r + 10^j; k:= k - 10^j mod n;
    fi
od;
if k = 1 then r:= r + 1 fi;
r
end proc:
seq(f(n),n=1..100); # Robert Israel, Feb 09 2016

a[n_] := For[k = 1, True, k++, b = FromDigits[ IntegerDigits[k, 2] ]; If[Mod[b, n] == 0, Return[b]]]; a[0] = 0; Table[a[n], {n, 0, 34}] (* Jean-François Alcover, Jun 14 2013, after Reinhard Zumkeller *)
With[{c=Rest[Union[FromDigits/@Flatten[Table[Tuples[{1,0},i],{i,10}], 1]]]}, Join[{0},Flatten[ Table[ Select[c,Divisible[#,n]&,1],{n,40}]]]] (* Harvey P. Dale, Dec 07 2013 *)

a(n) = {if( n==0, return (0)); my(m = n); while (vecmax(digits(m)) != 1, m+=n); m;} \\ Michel Marcus, Feb 09 2016, May 27 2020

apply( {A004290(n)=for(k=1,2^n,(t=fromdigits(binary(k)))%n||return(t))}, [1..44]) \\ M. F. Hasler, Mar 04 2025

def A004290(n):
    if n > 0:
        for i in range(1,2**n):
            x = int(bin(i)[2:])
            if not x % n:
                return x
    return 0
# Chai Wah Wu, Dec 30 2014

a(n) = n*A079339(n). - Jonathan Sondow, Jun 15 2014

Initial 0 deleted and offset corrected by N. J. A. Sloane, Jan 31 2024

A079339 Least k such that the decimal representation of k*n contains only 1's and 0's.

Original entry on oeis.org

1, 5, 37, 25, 2, 185, 143, 125, 12345679, 1, 1, 925, 77, 715, 74, 625, 653, 61728395, 579, 5, 481, 5, 4787, 4625, 4, 385, 40781893, 3575, 37969, 37, 3581, 3125, 3367, 3265, 286, 308641975, 3, 2895, 259, 25, 271, 2405, 25607, 25, 24691358, 23935, 213, 23125

Offset: 1

Benoit Cloitre, Feb 13 2003

From David Amar (dpamar(AT)gmail.com), Jul 12 2010: (Start)
This sequence is well defined.
In the n+1 first repunits (see A002275), there are at least 2 numbers that have the same value modulo n (pigeonhole principle).
The difference between those two numbers contains only 1's and 0's in decimal representation. (End)
This actually proves the stronger statement that there is always a multiple of the form 111...000 (Thm. 1 in Wu, 2014), cf. A244859 for these multiples and A244927 for the k-values. - M. F. Hasler, Mar 04 2025

			3*37 = 111 and no integer k < 37 has this property, hence a(3)=37.

Popular Computing (Calabasas, CA), Z-Sequences, Vol. 4 (No. 34, A pr 1976), pages PC36-4 to PC37-6, but there are many errors (cf. A257343, A257344).

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (terms 1..1999 from T. D. Noe, terms 2000..9998 from N. J. A. Sloane [based on A004290])
Chai Wah Wu, Pigeonholes and repunits, Amer. Math. Monthly, 121 (2014), 529-533.

Cf. A004290, A070189, A078241-A078248, A096681-A096688, A257343, A257344, A257345.

d(n,i)=floor(n/10^(i-1))-10*floor(n/10^i);
test(n)=sum(i=1,ceil(log(n)/log(10)),if(d(n,i)*(1-d(n,i)),1,0));
a(n)=if(n<0,0,s=1; while(test(n*s)>0,s++); s)

a(n) = A004290(n)/n.
a(n) < 10^(n+1) / (9n). - Charles R Greathouse IV, Jan 09 2012
a(n) <= A244927(n), with equality for n <= 6. - M. F. Hasler, Mar 04 2025

More terms from Vladeta Jovovic and Matthew Vandermast, Feb 14 2003

Definition simplified by Franklin T. Adams-Watters, Jan 09 2012

A078248 Smallest multiple of n using only digits 0 and 9.

Original entry on oeis.org

9, 90, 9, 900, 90, 90, 9009, 9000, 9, 90, 99, 900, 9009, 90090, 90, 90000, 99909, 90, 99009, 900, 9009, 990, 990909, 9000, 900, 90090, 999, 900900, 9909909, 90, 999099, 900000, 99, 999090, 90090, 900, 999, 990090, 9009, 9000, 99999, 90090, 9909909

Offset: 1

Amarnath Murthy, Nov 23 2002

a(n) = min{A097256(k): k > 0 and A097256(k) mod n = 0}. [Reinhard Zumkeller, Jan 10 2012]

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A004290, A078241-A078248, A079339, A096681-A096688.

a078248 n = head [x | x <- tail a097256_list, mod x n == 0]
-- Reinhard Zumkeller, Jan 10 2012

With[{t=Flatten[Table[FromDigits[Join[{9},#]]&/@Tuples[{0,9},n],{n,0,6}]]},Flatten[Table[Select[t,Divisible[#,i]&,1],{i,50}]]] (* Harvey P. Dale, May 31 2014 *)

More terms from Ray Chandler, Jul 12 2004

A096688 Least k such that decimal representation of k*n contains only digits 0 and 9.

Original entry on oeis.org

9, 45, 3, 225, 18, 15, 1287, 1125, 1, 9, 9, 75, 693, 6435, 6, 5625, 5877, 5, 5211, 45, 429, 45, 43083, 375, 36, 3465, 37, 32175, 341721, 3, 32229, 28125, 3, 29385, 2574, 25, 27, 26055, 231, 225, 2439, 2145, 230463, 225, 2, 215415, 1917, 1875, 202041, 18, 1959

Offset: 1

Ray Chandler, Jul 12 2004

Cf. A004290, A078241-A078248, A079339, A096681-A096688.

a(n) = A078248(n)/n.

A096681 Least k such that decimal representation of k*n contains only digits 0 and 2.

Original entry on oeis.org

2, 1, 74, 5, 4, 37, 286, 25, 24691358, 2, 2, 185, 154, 143, 148, 125, 1306, 12345679, 1158, 1, 962, 1, 9574, 925, 8, 77, 81563786, 715, 75938, 74, 7162, 625, 6734, 653, 572, 61728395, 6, 579, 518, 5, 542, 481, 51214, 5, 49382716, 4787, 426, 4625, 44898, 4

Offset: 1

Ray Chandler, Jul 12 2004

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Cf. A004290, A078241-A078248, A079339, A096682, A096683, A096684, A096685, A096686, A096687, A096688.

isok(n) = my(vd = vecsort(digits(n),,8)); (vd == [0,2]) || (vd == [2]);
a(n) = my(k=1); while(!isok(k*n), k++); k; \\ Michel Marcus, Sep 25 2016

def next02(n):
  s = str(n)
  if s > '2'*len(s): return int('2' + '0'*len(s))
  for i, c in enumerate(s):
    if c == '1': return int(s[:i] + '2' + '0'*(len(s)-i-1))
    elif s[i:] > '2'*(len(s)-i): return int(s[:i-1] + '2' + '0'*(len(s)-i))
def a(n):
  k = 1
  while set(str(k*n)) - set('02') != set(): k = max(k+1, next02(k*n)//n)
  return k
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Feb 01 2021

a(n) = A078241(n)/n.

A169965 Numbers whose decimal expansion contains only 0's and 2's.

Original entry on oeis.org

0, 2, 20, 22, 200, 202, 220, 222, 2000, 2002, 2020, 2022, 2200, 2202, 2220, 2222, 20000, 20002, 20020, 20022, 20200, 20202, 20220, 20222, 22000, 22002, 22020, 22022, 22200, 22202, 22220, 22222, 200000, 200002, 200020, 200022, 200200, 200202, 200220, 200222

Offset: 1

N. J. A. Sloane, Aug 07 2010

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to carryless arithmetic

Cf. A007088, A014263, A169964, A169966, A169967.

Cf. A078241, A204093, A204094, A204095, A097256.

a169965 n = a169965_list !! (n-1)
a169965_list = map (* 2) a007088_list
-- Reinhard Zumkeller, Jan 10 2012

Map[FromDigits,Tuples[{0,2},6]] (* Paolo Xausa, Oct 30 2023 *)

print1(0);for(d=1,5,for(n=2^(d-1),2^d-1,print1(", ");forstep(i=d-1,0,-1,print1((n>>i)%2*2)))) \\ Charles R Greathouse IV, Nov 16 2011

lista(N) = vector(N, i, fromdigits(binary(i-1)*2)); \\ Ruud H.G. van Tol, Oct 26 2024

a(n+1) = Sum_{k>=0} A030308(n,k)*A093136(k+1). - Philippe Deléham, Oct 16 2011

a(n) = 2 * A007088(n-1).

A078242 Smallest multiple of n using only digits 0 and 3.

Original entry on oeis.org

3, 30, 3, 300, 30, 30, 3003, 3000, 333, 30, 33, 300, 3003, 30030, 30, 30000, 33303, 3330, 33003, 300, 3003, 330, 330303, 3000, 300, 30030, 333333333, 300300, 3303303, 30, 333033, 300000, 33, 333030, 30030, 33300, 333, 330030, 3003, 3000, 33333

Offset: 1

Amarnath Murthy, Nov 23 2002

a(n) = min{A169966(k): k > 1 and A169966(k) mod n = 0}. - Reinhard Zumkeller, Jan 10 2012

Reinhard Zumkeller and Chai Wah Wu, Table of n, a(n) for n = 1..10000 First 1000 terms from Reinhard Zumkeller

Cf. A004290, A078241-A078248, A079339, A096681-A096688.

a078242 n = head [x | x <- tail a169966_list, mod x n == 0]
-- Reinhard Zumkeller, Jan 10 2012

With[{lst=Rest[FromDigits/@Tuples[{0,3},10]]},Table[SelectFirst[lst,Mod[#,n]==0&],{n,50}]] (* Harvey P. Dale, May 31 2025 *)

def A078242(n):
    if n > 0:
        for i in range(1,2**n):
            x = 3*int(bin(i)[2:])
            if not x % n:
                return x
    return 0 # Chai Wah Wu, Dec 31 2014

More terms from Ray Chandler, Jul 12 2004

A078243 Smallest multiple of n using only digits 0 and 4.

Original entry on oeis.org

4, 4, 444, 4, 40, 444, 4004, 40, 444444444, 40, 44, 444, 4004, 4004, 4440, 400, 44404, 444444444, 44004, 40, 40404, 44, 440404, 4440, 400, 4004, 4404444444, 4004, 4404404, 4440, 444044, 4000, 444444, 44404, 40040, 444444444, 444, 44004, 40404

Offset: 1

Amarnath Murthy, Nov 23 2002

a(n) = min{A169967(k): k > 1 and A169967(k) mod n = 0}. [Reinhard Zumkeller, Jan 10 2012]

Reinhard Zumkeller, Table of n, a(n) for n = 1..998

Cf. A004290, A078241-A078248, A079339, A096681-A096688.

a078243 n = head [x | x <- tail a169967_list, mod x n == 0]
-- Reinhard Zumkeller, Jan 10 2012

More terms from Ray Chandler, Jul 12 2004

A078244 Smallest multiple of n using only digits 0 and 5.

Original entry on oeis.org

5, 50, 555, 500, 5, 5550, 5005, 5000, 555555555, 50, 55, 55500, 5005, 50050, 555, 50000, 55505, 5555555550, 55005, 500, 50505, 550, 550505, 555000, 50, 50050, 5505555555, 500500, 5505505, 5550, 555055, 500000, 555555, 555050, 5005

Offset: 1

Amarnath Murthy, Nov 23 2002

a(n) = min{A169964(k): k > 1 and A169964(k) mod n = 0}. [Reinhard Zumkeller, Jan 10 2012]

Reinhard Zumkeller, Table of n, a(n) for n = 1..998

Cf. A004290, A078241-A078248, A079339, A096681-A096688.

a078244 n = head [x | x <- tail a169964_list, mod x n == 0]
-- Reinhard Zumkeller, Jan 10 2012

Module[{mlts=Rest[FromDigits/@Tuples[{0,5},12]]},Table[ SelectFirst[ mlts,Divisible[ #,n]&],{n,40}]] (* Harvey P. Dale, Aug 14 2021 *)

More terms from Ray Chandler, Jul 12 2004

A078245 Smallest multiple of n using only digits 0 and 6.

Original entry on oeis.org

6, 6, 6, 60, 60, 6, 6006, 600, 666, 60, 66, 60, 6006, 6006, 60, 6000, 66606, 666, 66006, 60, 6006, 66, 660606, 600, 600, 6006, 666666666, 60060, 6606606, 60, 666066, 60000, 66, 66606, 60060, 6660, 666, 66006, 6006, 600, 66666, 6006, 6606606, 660, 6660, 660606

Offset: 1

Amarnath Murthy, Nov 23 2002

a(n) = min{A204093(k): k > 0 and A204093(k) mod n = 0}. [Reinhard Zumkeller, Jan 10 2012]

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)

Cf. A004290, A078241-A078248, A079339, A096681-A096688.

a078245 n = head [x | x <- tail a204093_list, mod x n == 0]
-- Reinhard Zumkeller, Jan 10 2012

With[{c=Rest[FromDigits/@Tuples[{0,6},10]]},Table[SelectFirst[c,Divisible[ #,n]&],{n,50}]] (* The program uses the SelectFirst function from Mathematica version 10 *) (* Harvey P. Dale, Apr 15 2015 *)

def A204093(n): return int(bin(n)[2:].replace('1', '6'))
def a(n):
    k = 1
    while A204093(k)%n: k += 1
    return A204093(k)
print([a(n) for n in range(1, 47)]) # Michael S. Branicky, Jun 06 2021

More terms from Ray Chandler, Jul 12 2004

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A004290 Least positive multiple of n that when written in base 10 uses only 0's and 1's.

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A079339 Least k such that the decimal representation of k*n contains only 1's and 0's.

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A078248 Smallest multiple of n using only digits 0 and 9.

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A096688 Least k such that decimal representation of k*n contains only digits 0 and 9.

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A096681 Least k such that decimal representation of k*n contains only digits 0 and 2.

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A169965 Numbers whose decimal expansion contains only 0's and 2's.

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A078242 Smallest multiple of n using only digits 0 and 3.

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