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

1,2

Comments

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

References

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).

Links

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.

Formula

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

Example

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

Prog

(PARI) 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)

Crossrefs

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

Sequence in context: A174507 A119483 A157809 * A257343 A244927 A043075

Adjacent sequences: A079336 A079337 A079338 * A079340 A079341 A079342

Keyword

base,nonn

Author

Benoit Cloitre, Feb 13 2003

Extensions

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

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

Status

approved