A244859 -id:A244859 - cp's OEIS frontend

A079339 Least k such that the decimal representation of k*n contains only 1's and 0's.

1, 5, 37, 25, 2, 185, 143, 125, 12345679, 1, 1, 925, 77, 715, 74, 625, 653, 61728395, 579, 5, 481, 5, 4787, 4625, 4, 385, 40781893, 3575, 37969, 37, 3581, 3125, 3367, 3265, 286, 308641975, 3, 2895, 259, 25, 271, 2405, 25607, 25, 24691358, 23935, 213, 23125

Offset: 1

Benoit Cloitre, Feb 13 2003

From David Amar (dpamar(AT)gmail.com), Jul 12 2010: (Start)
This sequence is well defined.
In the n+1 first repunits (see A002275), there are at least 2 numbers that have the same value modulo n (pigeonhole principle).
The difference between those two numbers contains only 1's and 0's in decimal representation. (End)
This actually proves the stronger statement that there is always a multiple of the form 111...000 (Thm. 1 in Wu, 2014), cf. A244859 for these multiples and A244927 for the k-values. - M. F. Hasler, Mar 04 2025

			3*37 = 111 and no integer k < 37 has this property, hence a(3)=37.

Popular Computing (Calabasas, CA), Z-Sequences, Vol. 4 (No. 34, A pr 1976), pages PC36-4 to PC37-6, but there are many errors (cf. A257343, A257344).

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (terms 1..1999 from T. D. Noe, terms 2000..9998 from N. J. A. Sloane [based on A004290])
Chai Wah Wu, Pigeonholes and repunits, Amer. Math. Monthly, 121 (2014), 529-533.

Cf. A004290, A070189, A078241-A078248, A096681-A096688, A257343, A257344, A257345.

d(n,i)=floor(n/10^(i-1))-10*floor(n/10^i);
test(n)=sum(i=1,ceil(log(n)/log(10)),if(d(n,i)*(1-d(n,i)),1,0));
a(n)=if(n<0,0,s=1; while(test(n*s)>0,s++); s)

a(n) = A004290(n)/n.
a(n) < 10^(n+1) / (9n). - Charles R Greathouse IV, Jan 09 2012
a(n) <= A244927(n), with equality for n <= 6. - M. F. Hasler, Mar 04 2025

More terms from Vladeta Jovovic and Matthew Vandermast, Feb 14 2003

Definition simplified by Franklin T. Adams-Watters, Jan 09 2012

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

Original entry on oeis.org

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

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

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A079339 Least k such that the decimal representation of k*n contains only 1's and 0's.

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions