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A378865 a(n) is the smallest positive integer k such that n*k uses none of the digits of n, or 0 if no such k exists.

%I A378865 #13 Feb 16 2025 22:39:28
%S A378865 2,2,2,2,2,2,2,2,2,0,2,3,2,2,2,2,2,2,2,0,3,2,2,4,4,3,2,2,2,0,2,2,2,2,
%T A378865 2,2,3,2,2,0,2,4,2,2,2,2,4,2,12,0,4,2,2,2,2,2,2,2,2,0,4,5,3,2,2,2,2,3,
%U A378865 2,0,4,2,2,3,4,2,2,2,2,0,3,2,2,3
%N A378865 a(n) is the smallest positive integer k such that n*k uses none of the digits of n, or 0 if no such k exists.
%C A378865 It is observed that a(10*m) = 0 and a(100*m + 5) = 0 for all positive integers m.
%C A378865 If m is a repdigit number (A010785) that does not have the digit 9, then a(m) = 2 and if m = 99...9, with t 9's, then a(m) = 11...2, i.e., (t - 1) 1's followed by 2, since 99...9 * 11...2 equals (t - 1) 1's followed by (t - 1) 8's, where k = 11...2 is the smallest number with this property.  In other words, a(A002283(m)) = A047855(m), for all positive integers m.
%H A378865 Dominic McCarty, <a href="/A378865/b378865.txt">Table of n, a(n) for n = 1..1000</a> (synthesized from _Michel Marcus_'s data at A074157)
%F A378865 a(n) = A074157(n)/n.
%e A378865 a(12) = 3, since 12*1 = 12, 12*2 = 24 have digits in common with 12, while 12*3 = 36 does not.
%Y A378865 Cf. A074157, A010785, A002283, A047855.
%K A378865 nonn,base
%O A378865 1,1
%A A378865 _Gonzalo Martínez_, Dec 09 2024