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

%I A079339 #56 Mar 04 2025 11:47:40
%S A079339 1,5,37,25,2,185,143,125,12345679,1,1,925,77,715,74,625,653,61728395,
%T A079339 579,5,481,5,4787,4625,4,385,40781893,3575,37969,37,3581,3125,3367,
%U A079339 3265,286,308641975,3,2895,259,25,271,2405,25607,25,24691358,23935,213,23125
%N A079339 Least k such that the decimal representation of k*n contains only 1's and 0's.
%C A079339 From David Amar (dpamar(AT)gmail.com), Jul 12 2010: (Start)
%C A079339 This sequence is well defined.
%C A079339 In the n+1 first repunits (see A002275), there are at least 2 numbers that have the same value modulo n (pigeonhole principle).
%C A079339 The difference between those two numbers contains only 1's and 0's in decimal representation. (End)
%C A079339 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
%D A079339 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).
%H A079339 Robert G. Wilson v, <a href="/A079339/b079339.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1999 from T. D. Noe, terms 2000..9998 from N. J. A. Sloane [based on A004290])
%H A079339 Chai Wah Wu, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.121.06.529">Pigeonholes and repunits</a>, Amer. Math. Monthly, 121 (2014), 529-533.
%F A079339 a(n) = A004290(n)/n.
%F A079339 a(n) < 10^(n+1) / (9n). - _Charles R Greathouse IV_, Jan 09 2012
%F A079339 a(n) <= A244927(n), with equality for n <= 6. - _M. F. Hasler_, Mar 04 2025
%e A079339 3*37 = 111 and no integer k < 37 has this property, hence a(3)=37.
%o A079339 (PARI) d(n,i)=floor(n/10^(i-1))-10*floor(n/10^i);
%o A079339 test(n)=sum(i=1,ceil(log(n)/log(10)),if(d(n,i)*(1-d(n,i)),1,0));
%o A079339 a(n)=if(n<0,0,s=1; while(test(n*s)>0,s++); s)
%Y A079339 Cf. A004290, A070189, A078241-A078248, A096681-A096688, A257343, A257344, A257345.
%K A079339 base,nonn
%O A079339 1,2
%A A079339 _Benoit Cloitre_, Feb 13 2003
%E A079339 More terms from _Vladeta Jovovic_ and _Matthew Vandermast_, Feb 14 2003
%E A079339 Definition simplified by _Franklin T. Adams-Watters_, Jan 09 2012