A004290 -id:A004290 - cp's OEIS frontend

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.

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.

A244955 Smallest positive multiple of n whose base-4 representation contains only 0's and 1's.

Original entry on oeis.org

1, 4, 21, 4, 5, 84, 21, 16, 81, 20, 341, 84, 65, 84, 1365, 16, 17, 324, 1045, 20, 21, 1364, 69, 336, 325, 260, 81, 84, 261, 5460, 341, 64, 1089, 68, 1365, 324, 4181, 4180, 273, 80, 1025, 84, 5461, 1364, 5445, 276, 20821, 336, 1029, 1300, 4437, 260, 5141

Offset: 1

Eric M. Schmidt, Jul 09 2014

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Ed Pegg Jr., 'Binary' Puzzle
Eric M. Schmidt, Sage code to compute this sequence (use b=4)
Chai Wah Wu, Pigeonholes and repunits, Amer. Math. Monthly, 121 (2014), 529-533.

Cf. A004284 (written in base 4), A004290, A244954-A244960.

Module[{nn=10,b4},b4=Rest[FromDigits[#,4]&/@Tuples[{0,1},nn]];Table[SelectFirst[b4,Mod[ #,n]==0&],{n,60}]] (* Harvey P. Dale, Feb 01 2024 *)

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

A244958 Smallest positive multiple of n whose base-7 representation contains only 0's and 1's.

Original entry on oeis.org

1, 8, 57, 8, 50, 19608, 7, 8, 351, 50, 2409, 19608, 351, 56, 2745, 400, 16864, 134856, 57, 400, 399, 2794, 17158, 19608, 50, 19552, 351, 56, 137257, 120450, 16864, 2752, 2409, 16864, 350, 134856, 117993, 19608, 351, 400, 134849, 137256, 344, 16808, 2745

Offset: 1

Eric M. Schmidt, Jul 09 2014

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Ed Pegg Jr., 'Binary' Puzzle
Eric M. Schmidt, Sage code to compute this sequence (use b=7)
Chai Wah Wu, Pigeonholes and repunits, Amer. Math. Monthly, 121 (2014), 529-533.

Cf. A004287 (written in base 7), A004290, A244954-A244960.

With[{cl=Rest[{FromDigits[#,7],FromDigits[#]}&/@Tuples[{0,1},7]]},Table[SelectFirst[cl,Mod[ #[[1]],n]==0&],{n,50}]][[;;,1]] (* Harvey P. Dale, Feb 01 2024 *)

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

A244959 Smallest positive multiple of n whose base 8 representation contains only 0's and 1's.

Original entry on oeis.org

1, 8, 9, 8, 65, 72, 299593, 8, 9, 520, 4169, 72, 65, 2396744, 585, 64, 4097, 72, 513, 520, 17044041, 33352, 33281, 72, 266825, 520, 513, 2396744, 266249, 4680, 4681, 64, 32769, 32776, 16814665, 72, 262145, 4104, 585, 520, 32841, 136352328, 36937, 33352, 585

Offset: 1

Eric M. Schmidt, Jul 09 2014

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Ed Pegg Jr., 'Binary' Puzzle
Eric M. Schmidt, Sage code to compute this sequence (use b=8)
Chai Wah Wu, Pigeonholes and repunits, Amer. Math. Monthly, 121 (2014), 529-533.

Cf. A004288 (written in base 8), A004290, A244954-A244960.

Module[{nn=10,b8},b8=Rest[FromDigits[#,8]&/@Tuples[{0,1},nn]];Table[SelectFirst[ b8,Mod[#,n]==0&],{n,100}]] (* Harvey P. Dale, Feb 03 2024 *)

def A244959(n):
    if n > 0:
        for i in range(1,2**n):
            x = int(bin(i)[2:],8)
            if not x % n:
                return x
    return 0 # Chai Wah Wu, Dec 30 2014

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

A286820 a(n) = smallest positive multiple of n whose factorial base representation contains only 0's and 1's.

Original entry on oeis.org

1, 2, 3, 8, 25, 6, 7, 8, 9, 30, 33, 24, 26, 126, 30, 32, 153, 126, 152, 120, 126, 726, 5888, 24, 25, 26, 27, 728, 145, 30, 31, 32, 33, 5066, 840, 144, 5883, 152, 5070, 120, 123, 126, 129, 5192, 720, 5888, 752, 144, 147, 150, 153, 728, 848, 864, 46200, 728

Offset: 1

Rémy Sigrist, Jun 24 2017

All terms belong to A059590.
a(n) = n iff n belongs to A059590.
The sequence is well defined: for any n > 0: according to the pigeonhole principle, among the n+1 first repunits in factorial base (A007489), there must be two distinct terms equal modulo n; their absolute difference is a positive multiple of n, and contains only 0's and 1's in factorial base.
This sequence is to factorial base what A004290 is to decimal base.

			The first terms are:
n   a(n)     a(n) in factorial base
--  ----     ----------------------
1   1        1
2   2        1,0
3   3        1,1
4   8        1,1,0
5   25       1,0,0,1
6   6        1,0,0
7   7        1,0,1
8   8        1,1,0
9   9        1,1,1
10  30       1,1,0,0
11  33       1,1,1,1
12  24       1,0,0,0
13  26       1,0,1,0
14  126      1,0,1,0,0
15  30       1,1,0,0
16  32       1,1,1,0
17  153      1,1,1,1,1
18  126      1,0,1,0,0
19  152      1,1,1,1,0
20  120      1,0,0,0,0

Rémy Sigrist, Table of n, a(n) for n = 1..2000
Wikipedia, Factorial number system
Index entries for sequences related to factorial base representation

Cf. A004290, A007489, A059590, A284750.

isA059590(n) = my (r=2); while (n, if (n%r > 1, return (0), n\=r; r++)); return (1)
a(n) = forstep (m=n, oo, n, if (isA059590(m), return (m)))

A334914 Least positive multiple of n that when written in base 10 uses only 0's, 1's, 2's and 3's.

Original entry on oeis.org

1, 2, 3, 12, 10, 12, 21, 32, 333, 10, 11, 12, 13, 112, 30, 32, 102, 1332, 133, 20, 21, 22, 23, 120, 100, 130, 1323, 112, 203, 30, 31, 32, 33, 102, 210, 1332, 111, 1102, 312, 120, 123, 210, 301, 132, 3330, 230, 1222, 1200, 1323, 100, 102, 312, 212, 2322, 110

Offset: 1

Bernard Schott, May 16 2020

a(n) = n iff n is in A007090; there is no isolated fixed point because fixed points are always in patterns of 4 consecutive terms, and the first few patterns are (0,1,2,3), (10,11,12,13), (20,21,22,23), (30,31,32,33), (100,101,102,103) ...

			a(18) = 1332 because 1332 is the smallest multiple of 18 whose decimal digits are all 0, 1, 2 or 3.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A004290 (similar, with digits 0 and 1), A181060 (similar, with digits 0, 1 and 2).

f:= proc(n) local k;
  for k from 1 do if convert(convert(k*n,base,10),set) subset {0,1,2,3} then return k*n fi od
end proc:
map(f, [$1..100]); # Robert Israel, May 18 2020

Table[SelectFirst[Rest @ Flatten [FromDigits /@ Tuples[Range[0, 3], 4]], Divisible[#, n] &], {n, 1, 55}] (* Amiram Eldar, May 16 2020 *)

a(n) = my(k=1); while(vecmax(digits(k*n))>3, k++); k*n \\ Michel Marcus, May 17 2020

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

A244956 Smallest positive multiple of n whose base-5 representation contains only 0's and 1's.

Original entry on oeis.org

1, 6, 6, 156, 5, 6, 126, 656, 126, 30, 781, 156, 26, 126, 30, 656, 3281, 126, 3781, 780, 126, 3256, 3151, 97656, 25, 26, 756, 756, 15631, 30, 31, 3776, 16401, 3876, 630, 756, 3256, 3876, 156, 3280, 656, 126, 15781, 3256, 630, 16376, 3901, 472656, 15631, 150

Offset: 1

Eric M. Schmidt, Jul 09 2014

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Ed Pegg Jr., 'Binary' Puzzle
Eric M. Schmidt, Sage code to compute this sequence (use b=5)
Chai Wah Wu, Pigeonholes and repunits, Amer. Math. Monthly, 121 (2014), 529-533.

Cf. A004285 (written in base 5), A004290, A244954-A244960.

Module[{nn=10,b5},b5=Rest[FromDigits[#,5]&/@Tuples[{0,1},nn]];Table[SelectFirst[b5,Mod[ #,n]==0&],{n,100}]] (* Harvey P. Dale, Feb 01 2024 *)

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

A244957 Smallest positive multiple of n whose base-6 representation contains only 0's and 1's.

Original entry on oeis.org

1, 6, 6, 36, 1555, 6, 7, 216, 36, 9330, 253, 36, 1339, 42, 9330, 1296, 1513, 36, 7999, 55980, 42, 1518, 253, 216, 9325, 8034, 216, 252, 47995, 9330, 217, 7776, 1518, 9078, 12093235, 36, 37, 47994, 8034, 335880, 46699, 42, 43, 9108, 55980, 1518, 48175, 1296

Offset: 1

Eric M. Schmidt, Jul 09 2014

Eric M. Schmidt, Table of n, a(n) for n = 0..1000
Ed Pegg Jr., 'Binary' Puzzle
Eric M. Schmidt, Sage code to compute this sequence (use b=6)
Chai Wah Wu, Pigeonholes and repunits, Amer. Math. Monthly, 121 (2014), 529-533.

Cf. A004286 (written in base 6), A004290, A244954-A244960.

Module[{nn=10,b6},b6=Rest[FromDigits[#,6]&/@Tuples[{0,1},nn]];Table[SelectFirst[ b6,Mod[#,n]==0&],{n,100}]] (* Harvey P. Dale, Feb 03 2024 *)

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

A288538 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, n * a(n) does not contain the digit 1 in decimal base.

Original entry on oeis.org

2, 1, 3, 5, 4, 6, 7, 8, 10, 9, 19, 17, 16, 18, 15, 13, 12, 14, 11, 20, 22, 21, 23, 24, 25, 26, 27, 28, 30, 29, 32, 31, 62, 59, 58, 57, 55, 54, 52, 50, 49, 53, 48, 46, 45, 44, 47, 43, 41, 40, 56, 39, 42, 38, 37, 51, 36, 35, 34, 60, 63, 33, 61, 64, 65, 66, 67

Offset: 1

Rémy Sigrist, Jun 24 2017

This sequence is a permutation of the natural numbers; for any n > 0, there are infinitely many values v such that n * v does not contain the digit 1 in decimal base (for example: 2 * A004290(n) / n * 10^k for k = 0, 1, 2,...), hence n will eventually be chosen.
This sequence is self-inverse.
If n is a fixed point, then n^2 belongs to A052383.
The first fixed points are: 3, 6, 7, 8, 15, 20, 23, 24, 25, 26, 27, 28, 45, 47, 60, 64, 65, 66, 67, 68, 75, 76, 77, 78, ...

			The first terms, alongside n * a(n), are:
n       a(n)    n * a(n)
--      ----    --------
1       2       2
2       1       2
3       3       9
4       5       20
5       4       20
6       6       36
7       7       49
8       8       64
9       10      90
10      9       90
11      19      209
12      17      204
13      16      208
14      18      252
15      15      225
16      13      208
17      12      204
18      14      252
19      11      209
20      20      400

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A288538
Index entries for sequences that are permutations of the natural numbers
Rémy Sigrist, Logarithmic scatterplot of the first 10000 terms of n * a(n)

Cf. A004290, A052383.

cp's OEIS Frontend

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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A244955 Smallest positive multiple of n whose base-4 representation contains only 0's and 1's.

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A244958 Smallest positive multiple of n whose base-7 representation contains only 0's and 1's.

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A244959 Smallest positive multiple of n whose base 8 representation contains only 0's and 1's.

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A286820 a(n) = smallest positive multiple of n whose factorial base representation contains only 0's and 1's.

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A334914 Least positive multiple of n that when written in base 10 uses only 0's, 1's, 2's and 3's.

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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

A244956 Smallest positive multiple of n whose base-5 representation contains only 0's and 1's.

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Mathematica

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A244957 Smallest positive multiple of n whose base-6 representation contains only 0's and 1's.

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