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A305193
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Number of connected factorizations of n.
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13
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0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 4, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 5, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 7, 2, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 3, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 10, 1, 1, 2, 2, 1, 1, 1, 7, 5, 1, 1, 3, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 12, 1, 2, 2, 5, 1, 1, 1, 4, 1
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OFFSET
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1,4
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COMMENTS
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Given a finite multiset S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. This sequence counts factorizations S such that G(S) is a connected graph.
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LINKS
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EXAMPLE
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The a(72) = 10 factorizations:
(72),
(2*2*18), (2*3*12), (2*6*6), (3*4*6),
(2*36), (3*24), (4*18), (6*12),
(2*2*3*6).
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MATHEMATICA
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zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[Less@@#, GCD@@s[[#]]]>1&]}, If[c=={}, s, zsm[Union[Append[Delete[s, List/@c[[1]]], LCM@@s[[c[[1]]]]]]]]];
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], Length[zsm[#]]==1&]], {n, 100}]
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PROG
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(PARI)
is_connected(facs) = { my(siz=length(facs)); if(1==siz, 1, my(m=matrix(siz, siz, i, j, (gcd(facs[i], facs[j])!=1))^siz); for(n=1, siz, if(0==vecmin(m[n, ]), return(0))); (1)); };
A305193aux(n, m, facs) = if(1==n, is_connected(Set(facs)), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs, d); s += A305193aux(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Nov 07 2018
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CROSSREFS
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Cf. A048143, A281116, A286518, A286520, A290103, A303837, A304118, A304714, A304716, A305078, A305079, A319786.
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KEYWORD
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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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