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A339564
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Number of ways to choose a distinct factor in a factorization of n (pointed factorizations).
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25
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0, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 7, 1, 3, 3, 7, 1, 7, 1, 7, 3, 3, 1, 14, 2, 3, 4, 7, 1, 10, 1, 12, 3, 3, 3, 17, 1, 3, 3, 14, 1, 10, 1, 7, 7, 3, 1, 26, 2, 7, 3, 7, 1, 14, 3, 14, 3, 3, 1, 25, 1, 3, 7, 19, 3, 10, 1, 7, 3, 10, 1, 36, 1, 3, 7, 7, 3, 10, 1, 26, 7, 3
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OFFSET
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1,4
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LINKS
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FORMULA
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a(n) = Sum_{d|n, d>1} A001055(n/d).
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EXAMPLE
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The pointed factorizations of n for n = 2, 4, 6, 8, 12, 24, 30:
((2)) ((4)) ((6)) ((8)) ((12)) ((24)) ((30))
((2)*2) ((2)*3) ((2)*4) ((2)*6) ((3)*8) ((5)*6)
(2*(3)) (2*(4)) (2*(6)) (3*(8)) (5*(6))
((2)*2*2) ((3)*4) ((4)*6) ((2)*15)
(3*(4)) (4*(6)) (2*(15))
((2)*2*3) ((2)*12) ((3)*10)
(2*2*(3)) (2*(12)) (3*(10))
((2)*2*6) ((2)*3*5)
(2*2*(6)) (2*(3)*5)
((2)*3*4) (2*3*(5))
(2*(3)*4)
(2*3*(4))
((2)*2*2*3)
(2*2*2*(3))
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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[Union[fac]], {fac, facs[n]}], {n, 50}]
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CROSSREFS
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Choosing a position instead of value gives A066637.
The ordered additive version is A336875.
A001787 count normal multisets with a selected position.
A001792 counts compositions with a selected position.
A006128 counts partitions with a selected position.
A066186 count strongly normal multisets with a selected position.
A254577 counts ordered factorizations with a selected position.
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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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