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A067132
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Number of elements in the largest set of divisors of n which are in geometric progression.
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2
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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FORMULA
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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).
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EXAMPLE
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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}.
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MATHEMATICA
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a[n_] := If[n==1, 1, Max@@Last/@FactorInteger[n]+1]
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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