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

1,2

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

Links

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

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

Example

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}.

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]

Crossrefs

Cf. A000961, A067131, A051903.

Sequence in context: A375767 A375769 A375429 * A336500 A328401 A207666

Adjacent sequences: A067129 A067130 A067131 * A067133 A067134 A067135

Keyword

easy,nonn

Author

_Amarnath Murthy_, Jan 09 2002

Extensions

Edited by _Dean Hickerson_, Jan 15 2002

Status

approved