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A336569
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Number of maximal strict chains of divisors from n to 1 using elements of A130091 (numbers with distinct prime multiplicities).
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13
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1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 0, 1, 1, 2, 1, 2, 0, 0, 1, 3, 1, 0, 1, 2, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 3, 1, 0, 1, 2, 2, 0, 1, 4, 1, 2, 0, 2, 1, 3, 0, 3, 0, 0, 1, 0, 1, 0, 2, 1, 0, 0, 1, 2, 0, 0, 1, 5, 1, 0, 2, 2, 0, 0, 1, 4, 1, 0, 1, 0, 0, 0, 0
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OFFSET
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1,12
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COMMENTS
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A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.
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LINKS
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EXAMPLE
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The a(n) chains for n = 12, 72, 144, 192 (ones not shown):
12/3 72/18/2 144/72/18/2 192/96/48/24/12/3
12/4/2 72/18/9/3 144/72/18/9/3 192/64/32/16/8/4/2
72/24/12/3 144/48/24/12/3 192/96/32/16/8/4/2
72/24/8/4/2 144/72/24/12/3 192/96/48/16/8/4/2
72/24/12/4/2 144/48/16/8/4/2 192/96/48/24/8/4/2
144/48/24/8/4/2 192/96/48/24/12/4/2
144/72/24/8/4/2
144/48/24/12/4/2
144/72/24/12/4/2
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MATHEMATICA
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strsigQ[n_]:=UnsameQ@@Last/@FactorInteger[n];
fasmax[y_]:=Complement[y, Union@@(Most[Subsets[#]]&/@y)];
strchs[n_]:=If[n==1, {{}}, If[!strsigQ[n], {}, Join@@Table[Prepend[#, d]&/@strchs[d], {d, Select[Most[Divisors[n]], strsigQ]}]]];
Table[Length[fasmax[strchs[n]]], {n, 100}]
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CROSSREFS
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A336423 is the non-maximal version.
A336570 is the version for chains not necessarily containing n.
A001222 counts prime factors with multiplicity.
A007425 counts divisors of divisors.
A045778 counts strict factorizations.
A071625 counts distinct prime multiplicities.
A074206 counts strict chains of divisors from n to 1.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
Cf. A002033, A005117, A098859, A118914, A124010, A305149, A327498, A327523, A336414, A336425, A336500, A336568.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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