A004290 -id:A004290 - cp's OEIS frontend

A334963 a(n) is the least positive multiple of n that has at most two distinct digits.

1122, 515, 1144, 525, 212, 535, 1188, 545, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 600, 121, 122, 3444, 744, 500, 252, 889, 4224, 774, 10010, 131, 660, 133, 4422, 4455, 272, 411, 414, 556, 700, 141, 994, 858, 144, 3335, 292, 441, 444, 447, 300, 151, 2888

Offset: 102

David A. Corneth, May 17 2020

			a(102) = 1122 as 1122 = 11*102 is the least multiple of 102 that has at most 2 distinct digits.

David A. Corneth, Table of n, a(n) for n = 102..10101

Cf. A004290, A031955, A181060.

a[n_] := Module[{k = n}, While[Length @ Select[DigitCount[k, 10], # > 0 &] > 2, k += n]; k]; Array[a, 51, 102] (* Amiram Eldar, May 21 2020 *)

a(n) = for(i = 1, oo, if(#Set(digits(i*n))<3, return(i*n)))

a(n) <= A004290(n).

a(n) = n if n is in A031955. - Bernard Schott, May 17 2020

A370571 Smallest multiple of n that when written in base 10 uses only 0's and 1's and at least one of each.

Original entry on oeis.org

10, 10, 1011, 100, 10, 1110, 1001, 1000, 1011111111, 10, 110, 11100, 1001, 10010, 1110, 10000, 11101, 1111111110, 11001, 100, 10101, 110, 110101, 111000, 100, 10010, 1101111111, 100100, 1101101, 1110, 111011, 100000, 1101111, 111010, 10010, 11111111100, 1110, 110010, 10101, 1000

Offset: 1

Ivan N. Ianakiev, Feb 22 2024

For all n, a(n) exists (see proof in References).

Peter Winkler, Mathematical Puzzles (revised edition), CRC Press, 2024, p. liii.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A004290, A244859, A007088.

a[n_]:=Min[Select[FromDigits/@Tuples[{0,1},n+1],
Divisible[#,n]&&Union[IntegerDigits[#]]=={0,1}&]]; a/@Range[23]

from itertools import count
def a(n): return next(d for k in count(1) if ("0" in (b:=bin(k)[2:])) and (d:=int(b))%n==0)
print([a(n) for n in range(1, 24)]) # Michael S. Branicky, Feb 22 2024

a(10^e-1) <= 1^e 0 1^(8*e), where ^ denotes repeated concatenation of digits on the right-hand side. - Michael S. Branicky, Feb 22 2024

More terms from Michael S. Branicky, Feb 22 2024

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.

A334423 Fixed points of A257345.

Original entry on oeis.org

0, 1, 2, 4, 8, 16, 21, 32, 42, 64, 84, 128, 168, 256, 336, 512, 672, 1024, 1344, 2048, 2231, 2688, 4096, 4462, 5376, 8192, 9324, 10752, 16384, 18648, 21504, 32768, 37296, 43008, 65536, 74592, 86016, 131072, 149184, 172032, 262144, 298368, 344064, 524288, 596736, 688128, 1048576

Offset: 1

Bernard Schott, May 25 2020

The least positive multiple of an integer m that when written in base 10 uses only 0's and 1's is q = A004290(m) = k*m. If we regard q as binary number and converts q to base 10, we get A257345(q) = u. When m = u, then m is a term.
If m is a term, then m*2^k is another term.
The first 3 primitive terms are 1, 21, 2231 and the 3 corresponding subsequences of such fixed points are,
-> m = 0 or m = 2^k, k>=0 (A131577),
-> m = 21 * 2^k, k>=0 (A175805),
-> m = 2231 * 2^k, k>=0 (2231, 4462, 9324, 18648, ...).

			The least positive multiple of 42 that when written in base 10 uses only 0's and 1's is 101010 = 2405*42. If we regard 101010 as binary number and converts to base 10, we get 42; hence, 42 is a term.
Successive operations for first primitive terms:
1 --> A004290(1) = 1_{10} --> 1_{2} = 1_{10},
21 --> A004290(21) = 10101_{10} --> 10101_{2} = 21_{10},
2231 --> A004290(2231) = 100010110111_{10} --> 100010110111_{2} = 2231_{10}.

Cf. A004290, A079339, A257345.

Subsequences: A131577, A175805.

f(n) = {if( n==0, return (0)); my(m = n); while (vecmax(digits(m)) != 1, m+=n); m; } \\ A004290
isok(m) = fromdigits(digits(f(m), 10), 2) == m; \\ Michel Marcus, May 29 2020

A257345(A004290(a(n))) = a(n).

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A334963 a(n) is the least positive multiple of n that has at most two distinct digits.

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A370571 Smallest multiple of n that when written in base 10 uses only 0's and 1's and at least one of each.

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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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A334423 Fixed points of A257345.

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