A286518 -id:A286518 - cp's OEIS frontend

A305078 Heinz numbers of connected integer partitions.

2, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 37, 39, 41, 43, 47, 49, 53, 57, 59, 61, 63, 65, 67, 71, 73, 79, 81, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 117, 121, 125, 127, 129, 131, 133, 137, 139, 147, 149, 151, 157, 159, 163, 167

Offset: 1

Gus Wiseman, May 24 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

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 lists all Heinz numbers of multisets S such that G(S) is a connected graph.

			The sequence of all connected multiset multisystems (see A302242, A112798) begins:
   2: {{}}
   3: {{1}}
   5: {{2}}
   7: {{1,1}}
   9: {{1},{1}}
  11: {{3}}
  13: {{1,2}}
  17: {{4}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  29: {{1,3}}
  31: {{5}}
  37: {{1,1,2}}
  39: {{1},{1,2}}
  41: {{6}}
  43: {{1,4}}
  47: {{2,3}}
  49: {{1,1},{1,1}}
  53: {{1,1,1,1}}
  57: {{1},{1,1,1}}
  59: {{7}}
  61: {{1,2,2}}
  63: {{1},{1},{1,1}}
  65: {{2},{1,2}}
  67: {{8}}
  71: {{1,1,3}}
  73: {{2,4}}
  79: {{1,5}}
  81: {{1},{1},{1},{1}}
  83: {{9}}
  87: {{1},{1,3}}
  89: {{1,1,1,2}}
  91: {{1,1},{1,2}}
  97: {{3,3}}

Madeline Locus Dawsey, Tyler Russell and Dannie Urban, Polynomials Associated to Integer Partitions, arXiv:2108.00943 [math.NT], 2021.

Cf. A001221, A048143, A056239, A112798, A286518, A286520, A290103, A302242, A303837, A304118, A304714, A304716, A305052, A305055, A305079.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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]]]]]]]]];
Select[Range[300],Length[zsm[primeMS[#]]]==1&]

A304716 Number of integer partitions of n whose distinct parts are connected.

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 2, 6, 4, 9, 3, 15, 4, 18, 12, 25, 11, 41, 17, 54, 36, 72, 44, 113, 69, 145, 113, 204, 153, 302, 220, 394, 343, 541, 475, 771, 662, 1023, 968, 1398, 1314, 1929, 1822, 2566, 2565, 3440, 3446, 4677, 4688, 6187, 6407, 8216, 8544, 10975, 11436

Offset: 1

Gus Wiseman, May 17 2018

nonn

Given a finite set 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. For example, G({6,14,15,35}) is a 4-cycle. A multiset S is said to be connected if G(S) is a connected graph.

			The a(12) = 15 connected integer partitions and their corresponding connected multiset multisystems (see A112798, A302242) are the following.
                     (12): {{1,1,2}}
                    (6 6): {{1,2},{1,2}}
                    (8 4): {{1,1,1},{1,1}}
                    (9 3): {{2,2},{2}}
                   (10 2): {{1,3},{1}}
                  (4 4 4): {{1,1},{1,1},{1,1}}
                  (6 3 3): {{1,2},{2},{2}}
                  (6 4 2): {{1,2},{1,1},{1}}
                  (8 2 2): {{1,1,1},{1},{1}}
                (3 3 3 3): {{2},{2},{2},{2}}
                (4 4 2 2): {{1,1},{1,1},{1},{1}}
                (6 2 2 2): {{1,2},{1},{1},{1}}
              (4 2 2 2 2): {{1,1},{1},{1},{1},{1}}
            (2 2 2 2 2 2): {{1},{1},{1},{1},{1},{1}}
(1 1 1 1 1 1 1 1 1 1 1 1): {{},{},{},{},{},{},{},{},{},{},{},{}}

Cf. A000009, A003963, A048143, A054921, A218970, A285572, A286518, A302242, A304714, A305078, A305079, A322306, A322307.

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]]]]]]]]];
Table[Length[Select[IntegerPartitions[n],Length[zsm[Union[#]]]===1&]],{n,30}]

For n > 1, a(n) = A218970(n) + 1. - Gus Wiseman, Dec 04 2018

Name changed to distinguish from A218970 by Gus Wiseman, Dec 04 2018

A305079 Number of connected components of the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 2, 4, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 3, 1, 5, 2, 2, 2, 3, 1, 2, 1, 4, 1, 2, 1, 3, 2, 2, 1, 5, 1, 2, 2, 3, 1, 2, 2, 4, 1, 2, 1, 4, 1, 2, 1, 6, 1, 3, 1, 3, 2, 3, 1, 4, 1, 2, 2, 3, 2, 2, 1, 5, 1, 2, 1, 3, 2, 2, 1

Offset: 1

Gus Wiseman, May 24 2018

nonn

First differs from |A305052(n)| at a(169) = 1, A305052(169) = 0.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
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. If S is the integer partition with Heinz number n, a(n) is the number of connected components of G(S).

			The a(315) = 2 connected components of {2,2,3,4} are {{3},{2,2,4}}.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to Heinz numbers

Cf. A001221, A048143, A056239, A112798, A286518, A286520, A290103, A302242, A303837, A304118, A304714, A304716, A305052, A305055, A305078, A305501.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Table[Length[zsm[primeMS[n]]],{n,100}]

zero_first_elem_and_connected_elems(ys) = { my(cs = List([ys[1]]), i=1); ys[1] = 0; while(i<=#cs, for(j=2,#ys,if(ys[j]&&(1!=gcd(cs[i],ys[j])), listput(cs,ys[j]); ys[j] = 0)); i++); (ys); };
A007814(n) = valuation(n,2);
A000265(n) = (n/2^A007814(n));
A305079(n) = if(!(n%2),A007814(n)+A305079(A000265(n)), my(cs = apply(p -> primepi(p),factor(n)[,1]~), s=0); while(#cs, cs = select(c -> c, zero_first_elem_and_connected_elems(cs)); s++); (s)); \\ Antti Karttunen, Nov 10 2018

For all n, k > 0, we have a(2^n * k) = n + a(k).
For all x, y > 0, we have a(x * y) <= a(x) + a(y).
For x, y > 0 strongly coprime, we have a(x * y) = a(x) + a(y). Strongly coprime means every prime index of x is coprime to every prime index of y, where a prime index of n is a number m such that prime(m) divides n.
a(n) = A305501(A064989(n)) + A007814(n). - Antti Karttunen, Nov 10 2018

Terms and Mathematica program corrected by Gus Wiseman, Nov 10 2018

A304714 Number of connected strict integer partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 5, 2, 5, 5, 6, 5, 10, 6, 12, 12, 13, 14, 21, 17, 23, 26, 30, 31, 46, 38, 51, 55, 61, 70, 87, 85, 102, 116, 128, 138, 171, 169, 204, 225, 245, 272, 319, 334, 383, 429, 464, 515, 593, 629, 715, 790, 861, 950, 1082

Offset: 1

Gus Wiseman, May 17 2018

nonn

Given a finite set 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. For example, G({6,14,15,35}) is a 4-cycle. A multiset S is said to be connected if G(S) is a connected graph.

			The a(19) = 6 strict integer partitions are (19), (9,6,4), (10,5,4), (10,6,3), (12,4,3), (8,6,3,2). Taking the normalized prime factors of each part (see A112798, A302242), we have the following connected multiset multisystems.
       (19): {{8}}
    (9,6,4): {{2,2},{1,2},{1,1}}
   (10,5,4): {{1,3},{3},{1,1}}
   (10,6,3): {{1,3},{1,2},{2}}
   (12,4,3): {{1,1,2},{1,1},{2}}
  (8,6,3,2): {{1,1,1},{1,2},{2},{1}}

The Heinz numbers of these partitions are given by A328513.

Cf. A000009, A003963, A048143, A054921, A076078, A259936, A281116, A285572, A285573, A286518, A286520, A302242.

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]]]]]]]]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Length[zsm[#]]===1&]],{n,60}]

A286520 Number of finite connected sets of pairwise indivisible positive integers greater than one with least common multiple n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 5, 1, 1, 1, 1, 1, 5, 1, 1, 1, 3, 1, 5, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 17, 1, 1, 2, 1, 1, 5, 1, 2, 1, 5, 1, 9, 1, 1, 2, 2, 1, 5, 1, 4, 1, 1, 1, 17, 1, 1, 1

Offset: 2

Gus Wiseman, Jul 24 2017

nonn

Given a finite set 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 that are not relatively prime. For example, G({6,14,15,35}) is a 4-cycle. A set S is said to be connected if G(S) is a connected graph.

			The a(30)=5 sets are: {30}, {6,10}, {6,15}, {10,15}, {6,10,15}.

Cf. A048143, A054921, A076078, A259936, A281116, A285572, A285573, A286518.

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]]]]]]]]];
Table[Length[Select[Subsets[Rest[Divisors[n]]],And[!MemberQ[Tuples[#,2],{x_,y_}/;And[x

A303837 Number of z-trees with least common multiple n > 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 4, 1, 1, 1, 3, 1, 4, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 10, 1, 1, 2, 1, 1, 4, 1, 2, 1, 4, 1, 6, 1, 1, 2, 2, 1, 4, 1, 4, 1, 1, 1, 10, 1, 1, 1

Offset: 1

Gus Wiseman, May 19 2018

nonn

Given a finite set S of positive integers greater than 1, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices that have a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A set S is said to be connected if G(S) is a connected graph. The clutter density of S is defined to be Sum_{s in S} (omega(s) - 1) - omega(LCM(S)), where omega = A001221 and LCM is least common multiple. Then a z-tree is a finite connected set of pairwise indivisible positive integers greater than 1 with clutter density -1.
This is a generalization to multiset systems of the usual definition of hypertree (viz. connected hypergraph F such that two distinct hyperedges of F intersect in at most a common vertex and such that every cycle of F is contained in a hyperedge).
If n is squarefree with k prime factors, then a(n) = A030019(k).

			The a(72) = 6 z-trees together with the corresponding multiset systems (see A112798, A302242) are the following.
      (72): {{1,1,1,2,2}}
    (8,18): {{1,1,1},{1,2,2}}
    (8,36): {{1,1,1},{1,1,2,2}}
    (9,24): {{2,2},{1,1,1,2}}
   (6,8,9): {{1,2},{1,1,1},{2,2}}
  (8,9,12): {{1,1,1},{2,2},{1,1,2}}
The a(60) = 10 z-trees together with the corresponding multiset systems are the following.
       (60): {{1,1,2,3}}
     (4,30): {{1,1},{1,2,3}}
     (6,20): {{1,2},{1,1,3}}
    (10,12): {{1,3},{1,1,2}}
    (12,15): {{1,1,2},{2,3}}
    (12,20): {{1,1,2},{1,1,3}}
    (15,20): {{2,3},{1,1,3}}
   (4,6,10): {{1,1},{1,2},{1,3}}
   (4,6,15): {{1,1},{1,2},{2,3}}
  (4,10,15): {{1,1},{1,3},{2,3}}

Roland Bacher, On the enumeration of labelled hypertrees and of labelled bipartite trees, arXiv:1102.2708 [math.CO], 2011.

Cf. A006126, A030019, A048143, A076078, A112798, A275307, A285572, A286518, A286520, A293993, A293994, A302242, A303838, A304118.

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]]]]]]]]];
zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
Table[Length[Select[Rest[Subsets[Rest[Divisors[n]]]],And[zensity[#]==-1,zsm[#]=={n},Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]=={}]&]],{n,2,50}]

A305052 z-density of the integer partition with Heinz number n. Clutter density of the n-th multiset multisystem (A302242).

Original entry on oeis.org

0, -1, -1, -2, -1, -2, -1, -3, -1, -2, -1, -3, -1, -2, -2, -4, -1, -2, -1, -3, -1, -2, -1, -4, -1, -2, -1, -3, -1, -3, -1, -5, -2, -2, -2, -3, -1, -2, -1, -4, -1, -2, -1, -3, -2, -2, -1, -5, -1, -2, -2, -3, -1, -2, -2, -4, -1, -2, -1, -4, -1, -2, -1, -6, -1, -3

Offset: 1

Gus Wiseman, May 24 2018

sign

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The z-density of a multiset S of positive integers is Sum_{s in S} (omega(s) - 1) - omega(lcm(S)) where omega = A001221 is number of distinct prime factors.
First nonnegative entry after a(1) = 0 is a(169) = 0.

			The 1105th multiset multisystem is {{2},{1,2},{4}} with clutter density -2, so a(1105) = -2.
The 5429th multiset multisystem is {{1,2,2},{1,1,1,2}} with clutter density 0, so a(5429) = 0.
The 11837th multiset multisystem is {{1,1},{1,1,1},{1,1,1,2}} with clutter density -1, so a(11837) = -1.
The 42601th multiset multisystem is {{1,2},{1,3},{1,2,3}} with clutter density 1, so a(42601) = 1.

Cf. A001221, A030019, A048143, A056239, A112798, A285572, A286518, A286520, A290103, A302242, A303837, A304118, A304714, A304716, A304911, A305001.

zens[n_]:=If[n==1,0,Total@Cases[FactorInteger[n],{p_,k_}:>k*(PrimeNu[PrimePi[p]]-1)]-PrimeNu[LCM@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]]]];
Array[zens,100]

A304118 Number of z-blobs with least common multiple n > 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1

Offset: 1

Gus Wiseman, May 19 2018

nonn

Given a finite set S of positive integers greater than 1, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices that have a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A set S is said to be connected if G(S) is a connected graph. The clutter density of S is defined to be Sum_{s in S} (omega(s) - 1) - omega(LCM(S)), where omega = A001221 and LCM is least common multiple. A z-blob is a finite connected set S of pairwise indivisible positive integers greater than 1 such that no cap of S with at least two edges has clutter density -1.

If n is squarefree with k prime factors, then a(n) = A275307(k).

			The a(60) = 7 z-blobs together with the corresponding multiset systems (see A112798, A302242) are the following.
        (60): {{1,1,2,3}}
     (12,30): {{1,1,2},{1,2,3}}
     (20,30): {{1,1,3},{1,2,3}}
   (6,15,20): {{1,2},{2,3},{1,1,3}}
  (10,12,15): {{1,3},{1,1,2},{2,3}}
  (12,15,20): {{1,1,2},{2,3},{1,1,3}}
  (12,20,30): {{1,1,2},{1,1,3},{1,2,3}}
The a(120) = 14 z-blobs together with the corresponding multiset systems are the following.
       (120): {{1,1,1,2,3}}
     (24,30): {{1,1,1,2},{1,2,3}}
     (24,60): {{1,1,1,2},{1,1,2,3}}
     (30,40): {{1,2,3},{1,1,1,3}}
     (40,60): {{1,1,1,3},{1,1,2,3}}
   (6,15,40): {{1,2},{2,3},{1,1,1,3}}
  (10,15,24): {{1,3},{2,3},{1,1,1,2}}
  (12,15,40): {{1,1,2},{2,3},{1,1,1,3}}
  (12,30,40): {{1,1,2},{1,2,3},{1,1,1,3}}
  (15,20,24): {{2,3},{1,1,3},{1,1,1,2}}
  (15,24,40): {{2,3},{1,1,1,2},{1,1,1,3}}
  (20,24,30): {{1,1,3},{1,1,1,2},{1,2,3}}
  (24,30,40): {{1,1,1,2},{1,2,3},{1,1,1,3}}
  (24,40,60): {{1,1,1,2},{1,1,1,3},{1,1,2,3}}

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

Cf. A006126, A030019, A048143, A076078, A112798, A134954, A275307, A285572, A286518, A286520, A293993, A293994, A302242, A303837, A303838.

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]]]]]]]]];
zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
zreeQ[s_]:=And[Length[s]>=2,zensity[s]==-1];
zlobQ[s_]:=Apply[And,Composition[Not,zreeQ]/@Apply[LCM,zptns[s],{2}]];
zswell[s_]:=Union[LCM@@@Select[Subsets[s],Length[zsm[#]]==1&]];
zkernels[s_]:=Table[Select[s,Divisible[w,#]&],{w,zswell[s]}];
zptns[s_]:=Select[stableSets[zkernels[s],Length[Intersection[#1,#2]]>0&],Union@@#==s&];
stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
Table[If[n==1,0,Length[Select[Rest[Subsets[Rest[Divisors[n]]]],And[zsm[#]=={n},Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]=={},zlobQ[#]]&]]],{n,100}]

A327076 Maximum divisor of n that is 1 or connected.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 2, 9, 5, 11, 3, 13, 7, 5, 2, 17, 9, 19, 5, 21, 11, 23, 3, 25, 13, 27, 7, 29, 5, 31, 2, 11, 17, 7, 9, 37, 19, 39, 5, 41, 21, 43, 11, 9, 23, 47, 3, 49, 25, 17, 13, 53, 27, 11, 7, 57, 29, 59, 5, 61, 31, 63, 2, 65, 11, 67, 17, 23, 7, 71, 9, 73

Offset: 1

Gus Wiseman, Sep 05 2019

nonn

A number n with prime factorization n = prime(m_1)^s_1 * ... * prime(m_k)^s_k is connected if the simple labeled graph with vertex set {m_1,...,m_k} and edges between any two vertices with a common divisor greater than 1 is connected. Connected numbers are listed in A305078, which is the union of this sequence without 1.

Also the maximum MM-number (A302242) of a connected subset of the multiset of multisets with MM-number n.

Gus Wiseman, Sequences counting and encoding certain classes of multisets

Positions of prime numbers are A302569.
Connected numbers are A305078.
Cf. A007947, A056239, A112798, A286518, A302242, A304716, A305079, A322338, A322390, A322391.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
zsm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],GCD@@s[[#]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Table[Max[Select[Divisors[n],Length[zsm[primeMS[#]]]<=1&]],{n,30}]

If n is in A305078, then a(n) = n.

A303838 Number of z-forests with least common multiple n > 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 8, 1, 1, 2, 2, 2, 5, 1, 2, 2, 4, 1, 8, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 16, 1, 2, 3, 1, 2, 8, 1, 3, 2, 8, 1, 7, 1, 2, 3, 3, 2, 8, 1, 5, 1, 2, 1, 16, 2, 2

Offset: 1

Gus Wiseman, May 19 2018

nonn

Given a finite set S of positive integers greater than 1, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices that have a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A set S is said to be connected if G(S) is a connected graph. The clutter density of S is defined to be Sum_{s in S} (omega(s) - 1) - omega(LCM(S)), where omega = A001221 and LCM is least common multiple. A z-forest is a finite set of pairwise indivisible positive integers greater than 1 such that all connected components are z-trees, meaning they have clutter density -1.
This is a generalization to multiset systems of the usual definition of hyperforest (viz. hypergraph F such that two distinct hyperedges of F intersect in at most a common vertex and such that every cycle of F is contained in a hyperedge).
If n is squarefree with k prime factors, then a(n) = A134954(k).
Differs from A324837 at positions {1, 180, 210, ...}. For example, a(210) = 55, A324837(210) = 49.

			The a(60) = 16 z-forests together with the corresponding multiset systems (see A112798, A302242) are the following.
       (60): {{1,1,2,3}}
     (3,20): {{2},{1,1,3}}
     (4,15): {{1,1},{2,3}}
     (4,30): {{1,1},{1,2,3}}
     (5,12): {{3},{1,1,2}}
     (6,20): {{1,2},{1,1,3}}
    (10,12): {{1,3},{1,1,2}}
    (12,15): {{1,1,2},{2,3}}
    (12,20): {{1,1,2},{1,1,3}}
    (15,20): {{2,3},{1,1,3}}
    (3,4,5): {{2},{1,1},{3}}
   (3,4,10): {{2},{1,1},{1,3}}
    (4,5,6): {{1,1},{3},{1,2}}
   (4,6,10): {{1,1},{1,2},{1,3}}
   (4,6,15): {{1,1},{1,2},{2,3}}
  (4,10,15): {{1,1},{1,3},{2,3}}

Gus Wiseman, Table of n, a(n) for n = 1..250
Roland Bacher, On the enumeration of labelled hypertrees and of labelled bipartite trees, arXiv:1102.2708 [math.CO], 2011.

Cf. A006126, A030019, A048143, A076078, A112798, A134954, A275307, A285572, A286518, A286520, A293993, A293994, A302242, A303837, A304118.

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]]]]]]]]];
zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
Table[Length[Select[Rest[Subsets[Rest[Divisors[n]]]],Function[s,LCM@@s==n&&And@@Table[zensity[Select[s,Divisible[m,#]&]]==-1,{m,zsm[s]}]&&Select[Tuples[s,2],UnsameQ@@#&&Divisible@@#&]=={}]]],{n,100}]

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