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 A305193 Number of connected factorizations of n. 12
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS 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. a(n) depends only on prime signature of n (cf. A025487). - Antti Karttunen, Nov 07 2018 LINKS Antti Karttunen, Table of n, a(n) for n = 1..20736 Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000 EXAMPLE 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). MATHEMATICA 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}] PROG (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 A305193(n) = if(1==n, 0, A305193aux(n, n, List([]))); \\ Antti Karttunen, Nov 07 2018 CROSSREFS Cf. A048143, A281116, A286518, A286520, A290103, A303837, A304118, A304714, A304716, A305078, A305079, A319786. Sequence in context: A327658 A319786 A321271 * A038538 A293515 A326622 Adjacent sequences:  A305190 A305191 A305192 * A305194 A305195 A305196 KEYWORD nonn AUTHOR Gus Wiseman, May 27 2018 EXTENSIONS More terms from Antti Karttunen, Nov 07 2018 STATUS approved

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Last modified August 8 05:25 EDT 2020. Contains 336290 sequences. (Running on oeis4.)