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A342086
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Number of strict factorizations of divisors of n.
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15
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1, 2, 2, 3, 2, 5, 2, 5, 3, 5, 2, 9, 2, 5, 5, 7, 2, 9, 2, 9, 5, 5, 2, 16, 3, 5, 5, 9, 2, 15, 2, 10, 5, 5, 5, 18, 2, 5, 5, 16, 2, 15, 2, 9, 9, 5, 2, 25, 3, 9, 5, 9, 2, 16, 5, 16, 5, 5, 2, 31, 2, 5, 9, 14, 5, 15, 2, 9, 5, 15, 2, 34, 2, 5, 9, 9, 5, 15, 2, 25, 7, 5
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OFFSET
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1,2
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COMMENTS
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A strict factorization of n is a set of distinct positive integers > 1 with product n.
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LINKS
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EXAMPLE
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The a(1) = 1 through a(12) = 9 factorizations:
() () () () () () () () () () () ()
(2) (3) (2) (5) (2) (7) (2) (3) (2) (11) (2)
(4) (3) (4) (9) (5) (3)
(6) (8) (10) (4)
(2*3) (2*4) (2*5) (6)
(12)
(2*3)
(2*6)
(3*4)
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MAPLE
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sf1:= proc(n, m)
local D, d;
if n = 1 then return 1 fi;
D:= select(`<`, numtheory:-divisors(n) minus {1}, m);
add( procname(n/d, d), d= D)
end proc:
sf:= proc(n) option remember; sf1(n, n+1) end proc:f:= proc(n) local d; add(sf(d), d=numtheory:-divisors(n)) end proc:map(f, [$1..100]); # Robert Israel, Mar 10 2021
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MATHEMATICA
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facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Sum[Length[Select[facs[k], UnsameQ@@#&]], {k, Divisors[n]}], {n, 30}]
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CROSSREFS
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A version for partitions is A026906 (strict partitions of 1..n).
A version for partitions is A036469 (strict partitions of 0..n).
A version for partitions is A047966 (strict partitions of divisors).
A001222 counts prime-power divisors.
Cf. A001227, A050320, A340101, A340596, A340654, A340655, A340853, A341596, A341673, A341674, A342097.
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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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