A352219 a(n) is the least k such that A051037(n) | 60^k.

0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 3, 1, 3, 2, 2, 2, 2, 2, 3, 1, 3, 2, 2, 2, 4, 2, 3, 2, 3, 2, 3, 4, 3, 2, 2, 3, 4, 2, 3, 2, 3, 2, 2, 5, 3, 4, 3, 3, 2, 3, 4, 2, 3, 4, 2, 4, 3, 2, 3, 5, 3, 5, 3, 3, 2, 4, 4, 4, 3, 2, 6, 3, 4, 3, 4, 3, 2, 3, 5, 3, 5

Offset: 1

Author

Michael De Vlieger, Mar 08 2022

Keywords

• nonn
• easy
• base

Comments

Also, number of digits in the sexagesimal expansion of terminating unit fractions 1/A051037.

Examples

			a(1) = 0 since A051037(1) = 1 | 60^0.
a(2) = 1 since A051037(2) = 2 | 60^1; 1/2 in base 60 is represented by digit 30 after the radix point ":", i.e., :30.
a(7) = 2 since A051037(7) = 8 | 60^2; 1/8 in base 60 is :7,30, 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..10540 (A051037(10540) = 60^10)
• Eric Weisstein's World of Mathematics, Sexagesimal
• Wikipedia, Regular number.

Crossrefs

Cf. A051037, A086415, A117920.

Programs

Mathematica

With[{nn = 1024}, Sort[Flatten[Table[{2^a * 3^b * 5^c, Max[Ceiling[a/2], b, c]}, {a, 0, Log2[nn]}, {b, 0, Log[3, nn/(2^a)]}, {c, 0, Log[5, nn/(2^a*3^b)]}], 2]][[All, -1]] ]

Formula

a(n) ≍ n^(1/3), with lim sup a(n)/n^(1/3) being (6*log(2)*log(3)*log(5))^(1/3)/log(3) = 1.770... where A051037(n) is a power of 3 and the lim inf being (6*log(2)*log(3)*log(5))^(1/3)/log(60) = 0.4749... where A051037(n) is a power of 60. - Charles R Greathouse IV, Mar 08 2022