A079339 -id:A079339 - cp's OEIS frontend

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

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

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

Amarnath Murthy, Nov 23 2002

a(n) = min{A204094(k): k > 0 and A204094(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.

a078246 n = head [x | x <- tail a204094_list, mod x n == 0]
-- Reinhard Zumkeller, Jan 10 2012

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 *)

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

Amarnath Murthy, Nov 23 2002

a(n) = min{A204095(k): k > 0 and A204095(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.

a078247 n = head [x | x <- tail a204095_list, mod x n == 0]
-- Reinhard Zumkeller, Jan 10 2012

Module[{nn=10,lst},lst=Rest[FromDigits/@Tuples[{0,8},nn]];Table[SelectFirst[lst,Divisible[#,n]&],{n,50}]] (* Harvey P. Dale, Feb 20 2025 *)

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

More terms from Ray Chandler, Jul 12 2004

A096682 Least k such that decimal representation of k*n contains only digits 0 and 3.

Original entry on oeis.org

3, 15, 1, 75, 6, 5, 429, 375, 37, 3, 3, 25, 231, 2145, 2, 1875, 1959, 185, 1737, 15, 143, 15, 14361, 125, 12, 1155, 12345679, 10725, 113907, 1, 10743, 9375, 1, 9795, 858, 925, 9, 8685, 77, 75, 813, 715, 76821, 75, 74, 71805, 639, 625, 67347, 6, 653, 5775, 5661

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, A096681-A096688.

a(n) = A078242(n)/n.

A096686 Least k such that decimal representation of k*n contains only digits 0 and 7.

Original entry on oeis.org

7, 35, 259, 175, 14, 1295, 1, 875, 86419753, 7, 7, 6475, 539, 5, 518, 4375, 4571, 432098765, 4053, 35, 37, 35, 33509, 32375, 28, 2695, 285473251, 25, 265783, 259, 25067, 21875, 23569, 22855, 2, 2160493825, 21, 20265, 1813, 175, 1897, 185, 179249

Offset: 1

Ray Chandler, Jul 12 2004

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

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

f:= proc(n)  local q, q2, q5, n1, R, Agenda,d, newA, t, t1, t2;
q2:= padic:-ordp(n,2);
q5:= padic:-ordp(n,5);
q:= max(q2,q5);
n1:= n/2^q2/5^q5;
R[7]:= 7: Agenda:= {7}:
if 7 mod n1 = 0 then return 10^q*7/n fi;
for d from 2 do
    newA:= NULL;
    for t in Agenda do
      t1:= 10*t mod n1;
      if not assigned(R[t1]) then
        R[t1]:= 10*R[t];
        newA:= newA, t1;
      fi;
      t2:= (10*t+7) mod n1;
      if t2 = 0 then
        return 10^q*(10*R[t]+7)/n;
        break
      elif not assigned(R[t2]) then
        R[t2]:= 10*R[t]+7;
        newA:= newA,t2;
      fi;
    od;
    Agenda:= [newA];
od:
end proc:
map(f, [$1..50]); # Robert Israel, Mar 06 2017

f07[n_]:=Module[{k=1},While[!SubsetQ[{0,7},IntegerDigits[n*k]],k++];k]; Array[f07,8] (* The program generates the first 8 terms of the sequence. To generate more, increase the Array constant but because some of the terms are quite large the program may take a long time to run. *) (* Harvey P. Dale, Sep 25 2024 *)

a(n) = A078246(n)/n.

A096687 Least k such that decimal representation of k*n contains only digits 0 and 8.

Original entry on oeis.org

8, 4, 296, 2, 16, 148, 1144, 1, 98765432, 8, 8, 74, 616, 572, 592, 5, 5224, 49382716, 4632, 4, 3848, 4, 38296, 37, 32, 308, 326255144, 286, 303752, 296, 28648, 25, 26936, 2612, 2288, 24691358, 24, 2316, 2072, 2, 2168, 1924, 204856, 2, 197530864, 19148

Offset: 1

Ray Chandler, Jul 12 2004

Chai Wah Wu, Table of n, a(n) for n = 1..9998

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

def A096687(n):
    if n > 0:
        for i in range(1, 2**n):
            q, r = divmod(8*int(bin(i)[2:]), n)
            if not r:
                return q
    return 1 # Chai Wah Wu, Jan 02 2015

a(n) = A078247(n)/n.

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

A223474 Least positive multiple of n that when written in base 10 has digits in nonincreasing order.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 60, 52, 42, 30, 32, 51, 54, 76, 20, 21, 22, 92, 72, 50, 52, 54, 84, 87, 30, 31, 32, 33, 442, 70, 72, 74, 76, 663, 40, 41, 42, 43, 44, 90, 92, 94, 96, 98, 50, 51, 52, 53, 54, 55, 840, 741, 522, 531, 60, 61, 62, 63, 64, 65, 66, 871, 544, 552, 70, 71, 72, 73, 74, 75, 76, 77, 6552, 553, 80, 81, 82, 83, 84, 85

Offset: 1

Paul Tek, Mar 20 2013

This sequence is well defined (same reasoning as for A079339).

			a(39) = 663 because it is the least multiple of 39 appearing in A009996.

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

a(n)/n yields sequence A223475.
Cf. A004290, A181060.
Cf. A009996.

a[n_] := Block[{x=n}, While[0 < Max@Differences@IntegerDigits@x, x += n]; x]; Array[a, 85] (* Giovanni Resta, Mar 26 2013 *)

sub A223474 {
    my $n = shift;
    my $a = $n;
    while ($a !~ /^9*8*7*6*5*4*3*2*1*0*$/) {
        $a += $n;
    }
    return $a;
}
foreach (1..100) {
    print A223474($_), ",";
}

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 *)

cp's OEIS Frontend

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

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

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

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

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A096682 Least k such that decimal representation of k*n contains only digits 0 and 3.

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A096686 Least k such that decimal representation of k*n contains only digits 0 and 7.

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A096687 Least k such that decimal representation of k*n contains only digits 0 and 8.

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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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