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.

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A244954 Smallest positive multiple of n whose base-3 representation contains only 0's and 1's.

Original entry on oeis.org

1, 4, 3, 4, 10, 12, 28, 40, 9, 10, 121, 12, 13, 28, 30, 112, 85, 36, 247, 40, 84, 352, 253, 120, 325, 364, 27, 28, 841, 30, 31, 256, 363, 850, 280, 36, 37, 760, 39, 40, 82, 84, 3010, 352, 90, 1012, 94, 336, 2548, 850, 255, 364, 742, 108, 2200, 112, 741
Offset: 1

Views

Author

Eric M. Schmidt, Jul 09 2014

Keywords

Links

Crossrefs

Cf. A004283 (written in base 3), A004290, A244955-A244960.

Programs

  • Mathematica
    Module[{nn=10,b3},b3=Rest[FromDigits[#,3]&/@Tuples[{0,1},nn]];Table[SelectFirst[b3,Mod[ #,n]==0&],{n,60}]] (* Harvey P. Dale, Feb 01 2024 *)
  • PARI
    a(n,b=3) = if (n, k=1; while(vecmax(digits(k*n, b))>1, k++); k*n); \\ Michel Marcus, Jul 10 2014

Extensions

Data changed, offset changed, Mathematica program replaced, and b-file replaced by Harvey P. Dale, Feb 01 2024

A244960 Smallest positive multiple of n whose base 9 representation contains only 0's and 1's.

Original entry on oeis.org

1, 10, 9, 820, 10, 90, 91, 5380840, 9, 10, 7381, 7380, 91, 66430, 90, 43644592, 6562, 90, 7372, 820, 819, 65692, 7291, 48427560, 7300, 66430, 81, 532252, 59131, 90, 66340, 48368512, 66429, 6562, 66430, 7380, 532171, 7372, 819, 5380840, 82, 597870, 66349
Offset: 1

Views

Author

Eric M. Schmidt, Jul 09 2014

Keywords

Links

Crossrefs

Cf. A004289 (written in base 9), A004290, A244954-A244959.

Programs

  • Mathematica
    Module[{nn=15,b9},b9=Rest[FromDigits[#,9]&/@Tuples[{0,1},nn]];Table[SelectFirst[ b9,Mod[#,n]==0&],{n,100}]] (* Harvey P. Dale, Feb 03 2024 *)
  • PARI
    a(n,b=9) = if (n, k=1; while(vecmax(digits(k*n, b))>1, k++); k*n); \\ Michel Marcus, Jul 10 2014

Extensions

Data corrected, offset corrected, and b-file replaced by Harvey P. Dale, Feb 03 2024

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

Views

Author

Amarnath Murthy, Nov 23 2002

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Haskell
    a078242 n = head [x | x <- tail a169966_list, mod x n == 0]
-- Reinhard Zumkeller, Jan 10 2012
  • Mathematica
    With[{lst=Rest[FromDigits/@Tuples[{0,3},10]]},Table[SelectFirst[lst,Mod[#,n]==0&],{n,50}]] (* Harvey P. Dale, May 31 2025 *)
  • Python
    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

Extensions

More terms from Ray Chandler, Jul 12 2004

A190301 Smallest number h such that n*h is a repunit (A002275), or 0 if no such h exists.

Original entry on oeis.org

1, 0, 37, 0, 0, 0, 15873, 0, 12345679, 0, 1, 0, 8547, 0, 0, 0, 65359477124183, 0, 5847953216374269, 0, 5291, 0, 48309178743961352657, 0, 0, 0, 4115226337448559670781893, 0, 38314176245210727969348659, 0, 3584229390681, 0, 3367, 0, 0, 0, 3, 0, 2849, 0, 271, 0
Offset: 1

Views

Author

Jaroslav Krizek, May 07 2011

Keywords

Examples

			For n = 7: a(7) = 15873 because 7 * 15873 = 111111. Repunit 111111 is the smallest repunit with prime factor 7.

Links

Crossrefs

Cf. A084681 (repunit length), A216479 (the repunit).
Cf. A050782 = the smallest number h such that n*h is palindromic number, A083117 = the smallest number h such that n*h is repdigit number.
Cf. A004290, A079339, A181060, A181061.

Programs

  • Mathematica
    Table[If[GCD[n, 10] > 1, 0, k = MultiplicativeOrder[10, 9*n]; (10^k - 1)/(9*n)], {n, 100}] (* T. D. Noe, May 08 2011 *)
  • PARI
    a(n)=if(gcd(n,10)>1, 0, (10^znorder(Mod(10,9*n))-1)/9/n) \\ Charles R Greathouse IV, Aug 28 2016

A244859 Least positive multiple of n which when written in base 10 is either a repunit or of the form 111...000.

Original entry on oeis.org

0, 1, 10, 111, 100, 10, 1110, 111111, 1000, 111111111, 10, 11, 11100, 111111, 1111110, 1110, 10000, 1111111111111111, 1111111110, 111111111111111111, 100, 111111, 110, 1111111111111111111111, 111000, 100, 1111110, 111111111111111111111111111, 11111100
Offset: 0

Views

Author

Chai Wah Wu, Jul 07 2014

Keywords

Comments

a(1017) has 1008 digits. - Michael S. Branicky, Feb 22 2024
a(0) = 0 by convention: It can be considered as a repunit with zero digits, A002275(0) = (10^0-1)/9, and it is a positive multiple of n in the sense of k*n with k > 0. - M. F. Hasler, Mar 04 2025

Links

Crossrefs

Equal to A004290 for n = 1 .. 6.
Cf. A002275, A244927.

Programs

  • Maple
    A244859:= proc(n) local m,d2,d5;
d2:= padic:-ordp(n,2);
d5:= padic:-ordp(n,5);
m:= n/2^d2/5^d5;
10^max(d2,d5)*(10^numtheory:-order(10,9*m)-1)/9
end proc:
A244859(0):= 0:
seq(A244859(n),n= 0..100); # Robert Israel, Jul 08 2014
  • PARI
    apply( {A244859(n, m=Map(Mat([0,0])))=for(L=1,n, my(r=10^L\9); iferr(return(r-mapget(m,r%n)), E, mapput(m, r%n, r)))}, [0..33]) \\ M. F. Hasler, Mar 04 2025
  • Python
    def a(n):
    if n == 0: return 0
    moddict = {0: 0}
    for e in range(1, n+2):
        repe = (10**e-1)//9
        r = repe%n
        if r in moddict:
            return repe - moddict[r]
        else:
            moddict[r] = repe
print([a(n) for n in range(29)]) # Michael S. Branicky, Feb 22 2024

Formula

a(n) = n*A244927(n). - M. F. Hasler, Mar 04 2025
a(3^k) = (10^(3^k)-1)/9. For n > 0, A055642(a(n)) <= n. If n > 2 is not a power of 3, then A055642(a(n)) < n. - Chai Wah Wu, Mar 04 2025

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

Views

Author

Amarnath Murthy, Nov 23 2002

Keywords

Comments

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

Links

Crossrefs

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

Programs

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

Extensions

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

Views

Author

Amarnath Murthy, Nov 23 2002

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Haskell
    a078244 n = head [x | x <- tail a169964_list, mod x n == 0]
-- Reinhard Zumkeller, Jan 10 2012
  • Mathematica
    Module[{mlts=Rest[FromDigits/@Tuples[{0,5},12]]},Table[ SelectFirst[ mlts,Divisible[ #,n]&],{n,40}]] (* Harvey P. Dale, Aug 14 2021 *)

Extensions

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

Views

Author

Amarnath Murthy, Nov 23 2002

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Haskell
    a078245 n = head [x | x <- tail a204093_list, mod x n == 0]
-- Reinhard Zumkeller, Jan 10 2012
  • Mathematica
    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 *)
  • Python
    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

Extensions

More terms from Ray Chandler, Jul 12 2004

A078246 Smallest multiple of n using only digits 0 and 7.

Original entry on oeis.org

7, 70, 777, 700, 70, 7770, 7, 7000, 777777777, 70, 77, 77700, 7007, 70, 7770, 70000, 77707, 7777777770, 77007, 700, 777, 770, 770707, 777000, 700, 70070, 7707777777, 700, 7707707, 7770, 777077, 700000, 777777, 777070, 70, 77777777700, 777, 770070
Offset: 1

Views

Author

Amarnath Murthy, Nov 23 2002

Keywords

Comments

a(n) = min{A204094(k): k > 0 and A204094(k) mod n = 0}. [Reinhard Zumkeller, Jan 10 2012]

Links

Crossrefs

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

Programs

  • Haskell
    a078246 n = head [x | x <- tail a204094_list, mod x n == 0]
-- Reinhard Zumkeller, Jan 10 2012
  • Mathematica
    With[{c=Rest[FromDigits/@Tuples[{0,7},11]]},Table[SelectFirst[c, Divisible[ #,n]&],{n,40}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 27 2019 *)

Extensions

More terms from Ray Chandler, Jul 12 2004

A078247 Smallest multiple of n using only digits 0 and 8.

Original entry on oeis.org

8, 8, 888, 8, 80, 888, 8008, 8, 888888888, 80, 88, 888, 8008, 8008, 8880, 80, 88808, 888888888, 88008, 80, 80808, 88, 880808, 888, 800, 8008, 8808888888, 8008, 8808808, 8880, 888088, 800, 888888, 88808, 80080, 888888888, 888, 88008, 80808, 80, 88888, 80808
Offset: 1

Views

Author

Amarnath Murthy, Nov 23 2002

Keywords

Comments

a(n) = min{A204095(k): k > 0 and A204095(k) mod n = 0}. [Reinhard Zumkeller, Jan 10 2012]

Links

Crossrefs

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

Programs

  • Haskell
    a078247 n = head [x | x <- tail a204095_list, mod x n == 0]
-- Reinhard Zumkeller, Jan 10 2012
  • Mathematica
    Module[{nn=10,lst},lst=Rest[FromDigits/@Tuples[{0,8},nn]];Table[SelectFirst[lst,Divisible[#,n]&],{n,50}]] (* Harvey P. Dale, Feb 20 2025 *)
  • Python
    def a(n):
    k = 1
    while  8*int(bin(k)[2:])%n: k += 1
    return 8*int(bin(k)[2:])
print([a(n) for n in range(1, 43)]) # Michael S. Branicky, Aug 08 2021

Extensions

More terms from Ray Chandler, Jul 12 2004
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