A181061 -id:A181061 - cp's OEIS frontend

A078241 Smallest multiple of n using only digits 0 and 2.

2, 2, 222, 20, 20, 222, 2002, 200, 222222222, 20, 22, 2220, 2002, 2002, 2220, 2000, 22202, 222222222, 22002, 20, 20202, 22, 220202, 22200, 200, 2002, 2202222222, 20020, 2202202, 2220, 222022, 20000, 222222, 22202, 20020, 2222222220, 222

Offset: 1

Amarnath Murthy, Nov 23 2002

a(n) = min{A169965(k): k > 1 and A169965(k) mod n = 0}. - Reinhard Zumkeller, Jan 10 2012

Reinhard Zumkeller and Chai Wah Wu, Table of n, a(n) for n = 1..9998 First 998 terms from Reinhard Zumkeller.

Cf. A004290, A078242-A078248, A079339, A096681-A096688, A181060, A181061, A216481, A216812.

a078241 n = head [x | x <- tail a169965_list, mod x n == 0]
-- Reinhard Zumkeller, Jan 10 2012

Module[{m=Rest[FromDigits/@Tuples[{0,2},12]]},Table[Select[m,Divisible[ #,n]&,1],{n,40}]]//Flatten (* Harvey P. Dale, Jul 31 2017 *)

def A078241(n):
    if n > 0:
        for i in range(1,2**n):
            x = 2*int(bin(i)[2:])
            if not x % n:
                return x
    return 0 # Chai Wah Wu, Dec 30 2014

a(n) < 10^n / (0.45 n). - Charles R Greathouse IV, Jan 09 2012

a(n) <= A216812(n) <= 2(10^n - 1)/9. - N. J. A. Sloane, Sep 18 2012

More terms from Ray Chandler, Jul 12 2004

A181060 a(n) is the smallest positive multiple of n whose decimal digits are all 0, 1 or 2.

Original entry on oeis.org

1, 2, 12, 12, 10, 12, 21, 112, 12222, 10, 11, 12, 221, 112, 120, 112, 102, 12222, 1102, 20, 21, 22, 1012, 120, 100, 1222, 21222, 112, 1102, 120, 2201, 1120, 1122, 102, 210, 22212, 111, 1102, 10101, 120, 11111, 210, 2021, 220, 122220, 1012, 1222, 1200

Offset: 1

Herman Beeksma, Oct 01 2010

			a(9)=12222 because 12222 is the smallest multiple of 9 whose decimal digits are all 0, 1 or 2.

David A. Corneth, Table of n, a(n) for n = 1..10000
Inspired by Project Euler, Problem 303: Multiples with small digits.

a(n)/n yields sequence A181061.

With[{pms=Rest[Flatten[FromDigits/@Tuples[{0,1,2},6]]]},Table[ SelectFirst[ pms, Divisible[ #,n]&],{n,50}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 24 2017 *)

a(n) = my(k=1); while(vecmax(digits(k*n))>2, k++); k*n; \\ Michel Marcus, May 17 2020

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

Jaroslav Krizek, May 07 2011

nonn

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

Robert G. Wilson v, Table of n, a(n) for n = 1..1001 (first 300 terms from T. D. Noe)

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.

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 *)

a(n)=if(gcd(n,10)>1, 0, (10^znorder(Mod(10,9*n))-1)/9/n) \\ Charles R Greathouse IV, Aug 28 2016

A216479 a(n) is the least multiple of n which uses only the digit 1, or a(n) = -1 if no such multiple exists.

Original entry on oeis.org

1, -1, 111, -1, -1, -1, 111111, -1, 111111111, -1, 11, -1, 111111, -1, -1, -1, 1111111111111111, -1, 111111111111111111, -1, 111111, -1, 1111111111111111111111, -1, -1, -1, 111111111111111111111111111, -1, 1111111111111111111111111111, -1, 111111111111111, -1, 111111, -1, -1, -1, 111, -1, 111111, -1, 11111, -1

Offset: 1

V. Raman, Sep 07 2012

a(n) = -1 if and only if n is a multiple of 2 or 5. See comment in A216485. - Chai Wah Wu, Jun 21 2015

Chai Wah Wu, Table of n, a(n) for n = 1..1000

Cf. A084681 (number of 1's), A190301 (multiplier).

Cf. A004290, A079339, A181060, A181061.

Array[Which[GCD[#, 10] != 1, -1, Mod[#, 3] == 0, Block[{k = 1}, While[Mod[k, #] != 0, k = 10 k + 1]; k], True, (10^MultiplicativeOrder[10, #] - 1)/9] &, 42] (* Michael De Vlieger, Dec 11 2020 *)

def A216479(n):
    if n % 2 == 0 or n % 5 == 0:
        return -1
    rem = 1
    while rem % n != 0:
        rem = rem*10 + 1
    return rem
# Azanul Haque, Nov 28 2020

A223475 Least k such that the decimal representation of k*n has digits in nonincreasing order.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 4, 3, 2, 2, 3, 3, 4, 1, 1, 1, 4, 3, 2, 2, 2, 3, 3, 1, 1, 1, 1, 13, 2, 2, 2, 2, 17, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 15, 13, 9, 9, 1, 1, 1, 1, 1, 1, 1, 13, 8, 8, 1, 1, 1, 1, 1, 1, 1, 1, 84, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 86, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 5, 7, 5, 2, 5, 3, 4, 6, 1, 1, 75, 47, 38, 8, 45, 56, 8, 7, 5, 55, 5, 7

Offset: 1

Paul Tek, Mar 20 2013

			39*17 = 663 has digits in nonincreasing order, and no k < 17 has this property, hence a(39) = 17.

Giovanni Resta, Table of n, a(n) for n = 1..2000

a(n)*n yields sequence A223474.
Cf. A079339, A181061.
Cf. A009996.

a[n_] := a[nn_] := Block[{n = nn, f, w = Range@9, k = 1}, While[Mod[n, 10] == 0, n /= 10]; While[(f = Select[w, Max@ Differences@ IntegerDigits[n*#] <= 0 &, 1]) == {}, k++; w = Union@ Flatten@Table[ Select[d*10^(k-1) + w, Max@ Differences@ IntegerDigits[Mod[n*#, 10^k], 10, k] <= 0 &], {d, 0, 9}]]; f[[1]]]; Array[a, 123] (* faster than basic approach. Giovanni Resta, Mar 26 2013 *)

A216481 a(n) is the least multiple of n which uses only digit 2, or a(n) = -1 if no such multiple exists.

Original entry on oeis.org

2, 2, 222, -1, -1, 222, 222222, -1, 222222222, -1, 22, -1, 222222, 222222, -1, -1, 2222222222222222, 222222222, 222222222222222222, -1, 222222, 22, 2222222222222222222222, -1, -1, 222222, 222222222222222222222222222, -1, 2222222222222222222222222222, -1, 222222222222222, -1, 222222, 2222222222222222, -1, -1, 222, 222222222222222222, 222222, -1, 22222, 222222

Offset: 1

V. Raman, Sep 07 2012

Cf. A004290, A078241, A079339, A181060, A181061, A216812.

A216485 a(n) is the least value of k such that k*n uses only the digit 2, or a(n) = -1 if no such multiple exists.

Original entry on oeis.org

2, 1, 74, -1, -1, 37, 31746, -1, 24691358, -1, 2, -1, 17094, 15873, -1, -1, 130718954248366, 12345679, 11695906432748538, -1, 10582, 1, 96618357487922705314, -1, -1, 8547, 8230452674897119341563786, -1, 76628352490421455938697318, -1, 7168458781362, -1, 6734, 65359477124183, -1, -1, 6, 5847953216374269, 5698, -1, 542, 5291, 5167958656330749354

Offset: 1

V. Raman, Sep 07 2012

a(n) <= 2(10^n -1)/(9n). a(n) = -1 if and only if n is a multiple of 4 or 5. If n is a multiple of 4 then a(n) = -1 since 222....222 is not a multiple of 4. If n is a multiple of 5 then all multiples of n ends with the digit 0 or 5 and a(n) = -1. If n is odd and not a multiple of 4 or 5, then by the pigeonhole principle, two different repunits will have the same remainder modulo n. Their difference will be of the form 11...1110..0 which is a multiple of n. Since n and 10 are coprime, n is a divisor of a repunit and a(n) != -1. If n is even and not a multiple of 4 or 5, we take n/2 and use the same argument to show that n/2 is a divisor of a repunit and a(n) != -1. - Chai Wah Wu, Jun 21 2015

Chai Wah Wu, Table of n, a(n) for n = 1..1000

Cf. A004290, A079339, A181060, A181061.

A216482 a(n) is the least value of k such that k*n uses only digits 1 and 2. a(n) = -1 if no such multiple exists.

Original entry on oeis.org

1, 1, 4, 3, -1, 2, 3, 14, 1358, -1, 1, 1, 17, 8, -1, 7, 13, 679, 59, -1, 1, 1, 527, 88, -1, 47, 786, 4, 418, -1, 362, 66, 34, 33, -1, 617, 3, 319, 2849, -1, 271, 291, 284, 48, -1, 2657, 26, 44, 229, -1, 22, 406, 4, 393, -1, 2, 3723, 209, 19, -1, 2, 181, 194, 33, -1, 17, 33, 1634, 3219, -1, 172, 1696, 2907, 3, -1, 1462, 1443, 1554, 28, -1, 262, 271, 134, 1443

Offset: 1

V. Raman, Sep 07 2012

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A004290, A079339, A181060, A181061.

f:= proc(n) local d,a,i,S,R;
  if n mod 5 = 0 then return -1 fi;
  for d from ilog10(n)+1 do
     a:= (10^d-1)/9;
     S:= [seq(10^i, i=0..d-1)];
     R:= select(t -> convert(t,`+`) + a mod n = 0, combinat:-powerset(S));
     if R <> [] then return min(map(t -> convert(t,`+`)+a, R))/n fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, Dec 26 2022

A216478 a(n) is the least multiple of n which uses only digits 1 and 2. a(n) = -1 if no such multiple exists.

Original entry on oeis.org

1, 2, 12, 12, -1, 12, 21, 112, 12222, -1, 11, 12, 221, 112, -1, 112, 221, 12222, 1121, -1, 21, 22, 12121, 2112, -1, 1222, 21222, 112, 12122, -1, 11222, 2112, 1122, 1122, -1, 22212, 111, 12122, 111111, -1, 11111, 12222, 12212, 2112, -1, 122222, 1222, 2112, 11221, -1, 1122, 21112, 212, 21222, -1, 112, 212211, 12122, 1121

Offset: 1

V. Raman, Sep 07 2012

Cf. A004290, A079339, A181060, A181061.

A335401 a(n) is the smallest positive number such that the decimal digits of n*a(n) are all 0, 1, 2 or 3.

Original entry on oeis.org

1, 1, 1, 3, 2, 2, 3, 4, 37, 1, 1, 1, 1, 8, 2, 2, 6, 74, 7, 1, 1, 1, 1, 5, 4, 5, 49, 4, 7, 1, 1, 1, 1, 3, 6, 37, 3, 29, 8, 3, 3, 5, 7, 3, 74, 5, 26, 25, 27, 2, 2, 6, 4, 43, 2, 2, 23, 4, 17, 2, 2, 5, 21, 5, 2, 2, 3, 15, 19, 3, 3, 31, 14, 3, 4, 132, 3, 4, 27, 4, 41

Offset: 1

Bernard Schott, Jun 06 2020

If a(n) = k, then a(10n) = k.
a(n) = 1 iff n is in A007090; hence, except for a(1) = a(2) = a(3) = 1, the terms 1 always appear in strings of 4 consecutive 1's.
Records occur for n: 1, 4, 8, 9, 18, 76,  ...

			a(9)= 37 because 9*37=333 is the smallest multiple of 9 whose decimal digits are all 0, 1, 2 or 3.

Cf. A007090, A334914.

Cf. A079339 (similar, with digits 0 and 1), A181061 (similar, with digits 0, 1 and 2).

a[n_] := Block[{k = 1}, While[Max@ IntegerDigits[k n] > 3, k++]; k]; Array[a, 81] (* Giovanni Resta, Jun 06 2020 *)

a(n) = my(k=1); while(vecmax(digits(k*n))>3, k++); k; \\ Michel Marcus, Jun 08 2020

a(n) = A334914(n)/n.

More terms from Giovanni Resta, Jun 06 2020

cp's OEIS Frontend

A078241 Smallest multiple of n using only digits 0 and 2.

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A181060 a(n) is the smallest positive multiple of n whose decimal digits are all 0, 1 or 2.

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

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A216479 a(n) is the least multiple of n which uses only the digit 1, or a(n) = -1 if no such multiple exists.

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A223475 Least k such that the decimal representation of k*n has digits in nonincreasing order.

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A216481 a(n) is the least multiple of n which uses only digit 2, or a(n) = -1 if no such multiple exists.

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A216485 a(n) is the least value of k such that k*n uses only the digit 2, or a(n) = -1 if no such multiple exists.

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A216482 a(n) is the least value of k such that k*n uses only digits 1 and 2. a(n) = -1 if no such multiple exists.

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A216478 a(n) is the least multiple of n which uses only digits 1 and 2. a(n) = -1 if no such multiple exists.

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A335401 a(n) is the smallest positive number such that the decimal digits of n*a(n) are all 0, 1, 2 or 3.

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