A067132 Number of elements in the largest set of divisors of n which are in geometric progression.

1, 2, 2, 3, 2, 2, 2, 4, 3, 2, 2, 3, 2, 2, 2, 5, 2, 3, 2, 3, 2, 2, 2, 4, 3, 2, 4, 3, 2, 2, 2, 6, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 5, 3, 3, 2, 3, 2, 4, 2, 4, 2, 2, 2, 3, 2, 2, 3, 7, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 3, 3, 2, 2, 2, 5, 5, 2, 2, 3, 2, 2, 2, 4, 2, 3, 2, 3, 2, 2, 2, 6, 2, 3, 3, 3, 2, 2, 2

Offset: 1

Author

Amarnath Murthy, Jan 09 2002

Keywords

• easy
• nonn

Comments

Also a(n) = minimal 'freeness' of n with regard to squares, cubes, etc: All entries where a(n) = 2 are squarefree (or prime); Entries where a(n) = 3 are cubefree (and thus free of higher powers) but not squarefree, and so on. - Carl R. White, Jul 27 2009

For n > 1, a(n) is asymptotic to A000005(n)/A001221(n). - Eric Desbiaux, Dec 10 2012

Examples

			a(12) = 3 as the divisors of 12 are {1,2,3,4,6,12} and the maximal subsets in geometric progression are {1,2,4} and {3,6,12}.
a(16) = 5; the maximal set is {1,2,4,8,16}.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000961, A067131, A051903.

Programs

Maple

seq(max(0,seq(padic[ordp](n, p), p in numtheory[factorset](n))) + 1, n=1..100); # Ridouane Oudra, Sep 10 2024

Mathematica

a[n_] := If[n==1, 1, Max@@Last/@FactorInteger[n]+1]

Formula

If the prime factorization of n>1 is p_1^e_1 ... p_k^e_k, then a(n) = 1+max(e_1, ..., e_k).

a(n) = A051903(n) + 1. - Ridouane Oudra, Sep 10 2024

Extensions

Edited by Dean Hickerson, Jan 15 2002