%I #10 Mar 16 2022 19:59:51

%S 0,1,1,1,1,2,2,1,2,2,2,3,3,2,2,3,3,2,4,3,3,4,2,4,3,3,5,4,3,4,4,3,5,5,

%T 3,4,6,4,3,5,5,4,4,6,5,3,5,6,7,4,4,6,5,4,5,6,7,5,4,6,6,8,4,5,7,7,5,4,

%U 6,6,8,5,5,7,7,6,9,4,6,7,8,5,5,8,7,6,9

%N a(n) = least k such that A003586(n) | 12^k.

%C Also, number of digits in the duodecimal expansion of terminating unit fractions 1/A003586.

%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Chapter IX: The Representation of Numbers by Decimals, Theorem 136. 8th ed., Oxford Univ. Press, 2008, 144-145.

%H Michael De Vlieger, <a href="/A352072/b352072.txt">Table of n, a(n) for n = 1..10283</a> (A003586(10283) = 12^50)

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Duodecimal.html">Duodecimal</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Duodecimal">Duodecimal</a>.

%e a(1) = 0 since A003586(1) = 1 | 12^0.

%e a(2) = 1 since A003586(2) = 2 | 12^1; 1/2 expanded in base 12 = .6.

%e a(6) = 2 since A003586(6) = 8 | 12^2; 1/8 in base 12 = .16.

%e a(12) = 3 since A003586(12) = 27 | 12^3; 1/27 in base 12 = .054, etc.

%t With[{nn = 40000}, Sort[Join @@ Table[{2^a*3^b, Max[Ceiling[a/2], b]}, {a, 0, Log2[nn]}, {b, 0, Log[3, nn/(2^a)]}] ][[All, -1]] ]

%Y Cf. A003586, A086415, A117920.

%K nonn,easy,base

%O 1,6

%A _Michael De Vlieger_, Mar 08 2022