A299764 Number of special products of factorizations of n into factors > 1.
1, 2, 2, 5, 2, 6, 2, 10, 5, 6, 2, 16, 2, 6, 6, 18, 2, 16, 2, 16, 6, 6, 2, 36, 5, 6, 10, 16, 2, 22, 2, 32, 6, 6, 6, 44, 2, 6, 6, 36, 2, 22, 2, 16, 16, 6, 2, 72, 5, 16, 6, 16, 2, 36, 6, 36, 6, 6, 2, 64, 2, 6, 16, 51, 6, 22, 2, 16, 6, 22, 2, 104, 2, 6, 16, 16, 6
Offset: 1
Keywords
Examples
The a(12) = 16 special subset-products: 1<=(12), 12<=(12), 1<=(2*6), 2<=(2*6), 6<=(2*6), 12<=(2*6), 1<=(3*4), 3<=(3*4), 4<=(3*4), 12<=(3*4), 1<=(2*2*3), 2<=(2*2*3), 3<=(2*2*3), 4<=(2*2*3), 6<=(2*2*3), 12<=(2*2*3). The a(16) = 18 special subset-products: 1<=(16), 16<=(16), 1<=(4*4), 4<=(4*4), 16<=(4*4), 1<=(2*8), 2<=(2*8), 8<=(2*8), 16<=(2*8), 1<=(2*2*4), 2<=(2*2*4), 8<=(2*2*4), 16<=(2*2*4), 1<=(2*2*2*2), 2<=(2*2*2*2), 4<=(2*2*2*2), 8<=(2*2*2*2), 16<=(2*2*2*2).
Programs
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Mathematica
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]]; sppr[y_]:=Join@@Select[GatherBy[Union[Subsets[y]],Times@@#&],Length[#]===1&]; Table[Length[Join@@sppr/@facs[n]],{n,30}]
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