A079339 -id:A079339 - cp's OEIS frontend

A244927 Least positive number k such that k*n written in base 10 is either a repunit or of the form 1111....0000.

1, 5, 37, 25, 2, 185, 15873, 125, 12345679, 1, 1, 925, 8547, 79365, 74, 625, 65359477124183, 61728395, 5847953216374269, 5, 5291, 5, 48309178743961352657, 4625, 4, 42735, 4115226337448559670781893, 396825, 38314176245210727969348659, 37, 3584229390681, 3125, 3367, 326797385620915

Offset: 1

Chai Wah Wu, Jul 08 2014

			For n = 7, 15873*7 = 111111 and this is the least positive multiple of 7 that is either a repunit or of the form 1111...000.

M. F. Hasler, Table of n, a(n) for n = 1..1016 (first 100 terms from Chai Wah Wu), Mar 04 2025

Cf. A244859, A079339, A004290. Equal to A079339 for the first 6 terms.

a(n) = A244859(n)/n.

a(3^k) = (10^(3^k)-1)/3^(k+2). a(n) <= (10^n-1)/(9*n). If n > 2 is not a power of 3, then a(n) <= (10^(n-1)-1)/(9*n). - Chai Wah Wu, Mar 04 2025

A096683 Least k such that decimal representation of k*n contains only digits 0 and 4.

Original entry on oeis.org

4, 2, 148, 1, 8, 74, 572, 5, 49382716, 4, 4, 37, 308, 286, 296, 25, 2612, 24691358, 2316, 2, 1924, 2, 19148, 185, 16, 154, 163127572, 143, 151876, 148, 14324, 125, 13468, 1306, 1144, 12345679, 12, 1158, 1036, 1, 1084, 962, 102428, 1, 98765432

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, A096682, A096684, A096685, A096686, A096687, A096688.

f[n_] := Block[{id = {0, 4}, k = 1}, While[ Union[ Join[id, IntegerDigits[k*n]]] != id, k++]; k]; Array[f, 100] (* or *)
id = {0, 7}; lst = Union[ FromDigits /@ Flatten[ Table[ Tuples[id, j], {j, 15}], 1]]; If[ lst[[1]] == 0, lst = Rest@ lst]; f[n_] := (Min[ Select[lst, Mod[#, n] == 0 &]]/n) /. Infinity -> 0; Array[f, 100] (* or *)
id = {0, 7}; lst = Union[ FromDigits /@ Flatten[ Table[ Tuples[id, j], {j, 15}], 1]]; If[ lst[[1]] == 0, lst = Rest@ lst]; f[n_] := (SelectFirst[lst, Mod[#, n] == 0 &, 0]/n); a = Array[f, 100] (* requires Mathematica v10 *) (* Robert G. Wilson v, Sep 26 2016 *)

a(n) = A078243(n)/n.

A096684 Least k such that decimal representation of k*n contains only digits 0 and 5.

Original entry on oeis.org

5, 25, 185, 125, 1, 925, 715, 625, 61728395, 5, 5, 4625, 385, 3575, 37, 3125, 3265, 308641975, 2895, 25, 2405, 25, 23935, 23125, 2, 1925, 203909465, 17875, 189845, 185, 17905, 15625, 16835, 16325, 143, 1543209875, 15, 14475, 1295, 125, 1355

Offset: 1

Ray Chandler, Jul 12 2004

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

a(n) = A078244(n)/n.

A096685 Least k such that decimal representation of k*n contains only digits 0 and 6.

Original entry on oeis.org

6, 3, 2, 15, 12, 1, 858, 75, 74, 6, 6, 5, 462, 429, 4, 375, 3918, 37, 3474, 3, 286, 3, 28722, 25, 24, 231, 24691358, 2145, 227814, 2, 21486, 1875, 2, 1959, 1716, 185, 18, 1737, 154, 15, 1626, 143, 153642, 15, 148, 14361, 1278, 125, 134694, 12, 1306, 1155

Offset: 1

Ray Chandler, Jul 12 2004

k*n may comprise digits of 6 or both 0 and 6. - Harvey P. Dale, Dec 29 2013

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

k06[n_]:=Module[{k=1},While[Max[Drop[DigitCount[k*n],{6,10,4}]]>0,k++]; k]; Array[k06,52] (* Harvey P. Dale, Dec 29 2013 *)

a(n) = A078245(n)/n.

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

A257344 Erroneous version of A004290.

Original entry on oeis.org

1, 10, 111, 100, 10, 1110, 10101, 1000, 111111111, 10, 11, 11100, 1001, 10010, 1110, 10000, 11101, 1111111110, 100111, 100, 10101, 110, 1011011, 111000

Offset: 1

N. J. A. Sloane, Apr 29 2015

dead

Included in accordance with OEIS policy of including published but erroneous sequences so as to give links to the correct versions.

Popular Computing (Calabasas, CA), Z-Sequences, Vol. 4 (No. 34, Apr 1976), pages PC36-4 to PC37-6, but there are many errors (cf. A079339, A004290).

Cf. A079339, A004290.

A215160 Odd numbers n with the property that the binary representation of n is the same as the decimal representation of the smallest multiple of n that can be represented with only 1's and 0's.

Original entry on oeis.org

1, 21, 2231, 28261, 611123, 1200341, 3427673, 2202416417, 11102657671

Offset: 1

Patrick McKinley, Aug 05 2012

All numbers that are a power of 2 times a member of the sequence share the property that the binary representation is the same as the decimal representation of the first 1's and 0's multiple.

Of the values listed, only 1200341 and 3427673 are primes. - Jonathan Vos Post, Aug 09 2012

			For example 21*481=10101 (the first multiple of 21 containing only 1's and 0's) and the binary representation of 21 is 10101.

Cf. A079339.

rebase := proc(n,bin,bout)
    local a,c,i;
    a := 0 ;
    c := convert(n,base,bin) ;
    add( op(i,c)*bout^(i-1),i=1..nops(c)) ;
end proc:
isA079339 := proc(n,c)
    local c2,b;
    if modp(c,n) > 0 then
        return false;
    end if;
    c2 := rebase(c,10,2) ;
    for b from 1 to c2-1 do
        if modp( rebase(b,2,10),n) = 0 then
            return false;
        end if;
    end do:
    return true ;
end proc:
for n from 1 by 2 do
    sb := rebase(n,2,10) ;
    if isA079339(n,sb) then
        print(n);
    end if;
end do: # R. J. Mathar, Aug 09 2012

{odd n: n*A079339(n) = A007088(n)} . - R. J. Mathar, Aug 09 2012

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.

cp's OEIS Frontend

A244927 Least positive number k such that k*n written in base 10 is either a repunit or of the form 1111....0000.

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A096683 Least k such that decimal representation of k*n contains only digits 0 and 4.

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Mathematica

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A096684 Least k such that decimal representation of k*n contains only digits 0 and 5.

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A096685 Least k such that decimal representation of k*n contains only digits 0 and 6.

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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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Maple

A257344 Erroneous version of A004290.

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A215160 Odd numbers n with the property that the binary representation of n is the same as the decimal representation of the smallest multiple of n that can be represented with only 1's and 0's.

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Maple

Formula

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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