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Smallest multiple of n with digit sum = 2, or 0 if no such number exists, e.g., a(3k)=0.
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%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