A352072 a(n) = least k such that A003586(n) | 12^k.

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, 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, 6, 6, 8, 5, 5, 7, 7, 6, 9, 4, 6, 7, 8, 5, 5, 8, 7, 6, 9

Offset: 1

Author

Michael De Vlieger, Mar 08 2022

Keywords

• nonn
• easy
• base

Comments

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

Examples

			a(1) = 0 since A003586(1) = 1 | 12^0.
a(2) = 1 since A003586(2) = 2 | 12^1; 1/2 expanded in base 12 = .6.
a(6) = 2 since A003586(6) = 8 | 12^2; 1/8 in base 12 = .16.
a(12) = 3 since A003586(12) = 27 | 12^3; 1/27 in base 12 = .054, etc.
		

References

• 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.

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10283 (A003586(10283) = 12^50)
• Eric Weisstein's World of Mathematics, Duodecimal.
• Wikipedia, Duodecimal.

Crossrefs

Cf. A003586, A086415, A117920.

Programs

Mathematica

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]] ]