%I #14 Feb 12 2023 10:34:24
%S 2,2,0,20,20,0,1001,200,0,20,11,0,1001,10010,0,2000,100000001,0,
%T 1000000001,20,0,110,100000000001,0,200,10010,0,100100,
%U 100000000000001,0,0,20000,0,1000000010,10010,0,0,10000000010,0,200,0,0,0,1100,0
%N Smallest multiple of n with digit sum = 2, or 0 if no such number exists, e.g., a(3k)=0.
%C a(n)=0 if n is a multiple of 3, 31, 37, 41, 43, 53, 67, 71, 79, or 83. - _Ray Chandler_, Jul 30 2003
%H Robert Israel, <a href="/A069521/b069521.txt">Table of n, a(n) for n = 1..2016</a>
%F From _Robert Israel_, Feb 11 2023: (Start)
%F If n = 2^a * 5^b, a(n) = 2*10^max(a-1,b).
%F Otherwise, if n = 2^a*5^b*c where c is in A043292, then a(n) = 10^max(a,b) * (1 + 10^A069531(c)).
%F Otherwise a(n) = 0. (End)
%e a(7) = a(13) = 1001. Digit sum of 1001 = 2 and is the smallest such multiple of 7 and 13. a(17) = 100000001 = 17*5882353.
%p f:= proc(n) local a,b,c,k;
%p a:= padic:-ordp(n,2);
%p b:= padic:-ordp(n,5);
%p c:= n/(2^a*5^b);
%p if c = 1 then return 2*10^max(a-1,b) fi;
%p k:= traperror(NumberTheory:-ModularLog(-1,10,c));
%p if k = "no solutions exist" then 0
%p else 10^max(a,b) * (1 + 10^k)
%p fi
%p end proc:
%p map(f, [$1..50]); # _Robert Israel_, Feb 11 2023
%Y Cf. A043292, A069531.
%K base,nonn
%O 1,1
%A _Amarnath Murthy_, Apr 01 2002
%E More terms from _Ray Chandler_, Jul 30 2003