A304382 -id:A304382 - cp's OEIS frontend

A322110 Number of non-isomorphic connected multiset partitions of weight n that cannot be capped by a tree.

1, 1, 3, 6, 15, 32, 86, 216, 628, 1836, 5822

Offset: 0

Gus Wiseman, Nov 26 2018

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

The density of a multiset partition is defined to be the sum of numbers of distinct elements in each part, minus the number of parts, minus the total number of distinct elements in the whole partition. A multiset partition is a tree if it has more than one part, is connected, and has density -1. A cap is a certain kind of non-transitive coarsening of a multiset partition. For example, the four caps of {{1,1},{1,2},{2,2}} are {{1,1},{1,2},{2,2}}, {{1,1},{1,2,2}}, {{1,1,2},{2,2}}, {{1,1,2,2}}. - Gus Wiseman, Feb 05 2021

			The multiset partition C = {{1,1},{1,2,3},{2,3,3}} is not a tree but has the cap {{1,1},{1,2,3,3}} which is a tree, so C is not counted under a(8).
Non-isomorphic representatives of the a(1) = 1 through a(5) = 32 multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}        {{1,1,1,1,1}}
         {{1,2}}    {{1,2,2}}      {{1,1,2,2}}        {{1,1,2,2,2}}
         {{1},{1}}  {{1,2,3}}      {{1,2,2,2}}        {{1,2,2,2,2}}
                    {{1},{1,1}}    {{1,2,3,3}}        {{1,2,2,3,3}}
                    {{2},{1,2}}    {{1,2,3,4}}        {{1,2,3,3,3}}
                    {{1},{1},{1}}  {{1},{1,1,1}}      {{1,2,3,4,4}}
                                   {{1,1},{1,1}}      {{1,2,3,4,5}}
                                   {{1},{1,2,2}}      {{1},{1,1,1,1}}
                                   {{1,2},{1,2}}      {{1,1},{1,1,1}}
                                   {{2},{1,2,2}}      {{1},{1,2,2,2}}
                                   {{3},{1,2,3}}      {{1,2},{1,2,2}}
                                   {{1},{1},{1,1}}    {{2},{1,1,2,2}}
                                   {{1},{2},{1,2}}    {{2},{1,2,2,2}}
                                   {{2},{2},{1,2}}    {{2},{1,2,3,3}}
                                   {{1},{1},{1},{1}}  {{2,2},{1,2,2}}
                                                      {{2,3},{1,2,3}}
                                                      {{3},{1,2,3,3}}
                                                      {{4},{1,2,3,4}}
                                                      {{1},{1},{1,1,1}}
                                                      {{1},{1,1},{1,1}}
                                                      {{1},{1},{1,2,2}}
                                                      {{1},{2},{1,2,2}}
                                                      {{2},{1,2},{1,2}}
                                                      {{2},{1,2},{2,2}}
                                                      {{2},{2},{1,2,2}}
                                                      {{2},{3},{1,2,3}}
                                                      {{3},{1,3},{2,3}}
                                                      {{3},{3},{1,2,3}}
                                                      {{1},{1},{1},{1,1}}
                                                      {{1},{2},{2},{1,2}}
                                                      {{2},{2},{2},{1,2}}
                                                      {{1},{1},{1},{1},{1}}

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

Non-isomorphic tree multiset partitions are counted by A321229.
The weak-antichain case is counted by A322117.
The case without singletons is counted by A322118.
Cf. A002218, A007718, A013922, A030019, A056156, A275307, A304118, A304382, A304887, A305052, A305079, A321194.

Corrected by Gus Wiseman, Jan 27 2021

A322118 Number of non-isomorphic connected multiset partitions of weight n with no singletons that cannot be capped by a tree.

Original entry on oeis.org

1, 1, 2, 3, 7, 11, 29, 55, 155, 386, 1171

Offset: 0

Gus Wiseman, Nov 26 2018

The density of a multiset partition is defined to be the sum of numbers of distinct elements in each part, minus the number of parts, minus the total number of distinct elements in the whole partition. A multiset partition is a tree if it has more than one part, is connected, and has density -1. A cap is a certain kind of non-transitive coarsening of a multiset partition. For example, the four caps of {{1,1},{1,2},{2,2}} are {{1,1},{1,2},{2,2}}, {{1,1},{1,2,2}}, {{1,1,2},{2,2}}, {{1,1,2,2}}. - Gus Wiseman, Feb 05 2021

			The multiset partition C = {{1,1},{1,2,3},{2,3,3}} is not a tree but has the cap {{1,1},{1,2,3,3}} which is a tree, so C is not counted under a(8).
Non-isomorphic representatives of the a(2) = 2 through a(6) = 29 multiset partitions:
  {{1,1}}  {{1,1,1}}  {{1,1,1,1}}    {{1,1,1,1,1}}    {{1,1,1,1,1,1}}
  {{1,2}}  {{1,2,2}}  {{1,1,2,2}}    {{1,1,2,2,2}}    {{1,1,1,2,2,2}}
           {{1,2,3}}  {{1,2,2,2}}    {{1,2,2,2,2}}    {{1,1,2,2,2,2}}
                      {{1,2,3,3}}    {{1,2,2,3,3}}    {{1,1,2,2,3,3}}
                      {{1,2,3,4}}    {{1,2,3,3,3}}    {{1,2,2,2,2,2}}
                      {{1,1},{1,1}}  {{1,2,3,4,4}}    {{1,2,2,3,3,3}}
                      {{1,2},{1,2}}  {{1,2,3,4,5}}    {{1,2,3,3,3,3}}
                                     {{1,1},{1,1,1}}  {{1,2,3,3,4,4}}
                                     {{1,2},{1,2,2}}  {{1,2,3,4,4,4}}
                                     {{2,2},{1,2,2}}  {{1,2,3,4,5,5}}
                                     {{2,3},{1,2,3}}  {{1,2,3,4,5,6}}
                                                      {{1,1},{1,1,1,1}}
                                                      {{1,1,1},{1,1,1}}
                                                      {{1,1,2},{1,2,2}}
                                                      {{1,2},{1,1,2,2}}
                                                      {{1,2},{1,2,2,2}}
                                                      {{1,2},{1,2,3,3}}
                                                      {{1,2,2},{1,2,2}}
                                                      {{1,2,3},{1,2,3}}
                                                      {{1,2,3},{2,3,3}}
                                                      {{1,3,4},{2,3,4}}
                                                      {{2,2},{1,1,2,2}}
                                                      {{2,2},{1,2,2,2}}
                                                      {{2,3},{1,2,3,3}}
                                                      {{3,3},{1,2,3,3}}
                                                      {{3,4},{1,2,3,4}}
                                                      {{1,1},{1,1},{1,1}}
                                                      {{1,2},{1,2},{1,2}}
                                                      {{1,2},{1,3},{2,3}}

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

Non-isomorphic tree multiset partitions are counted by A321229, or A321231 without singletons.
The version with singletons is A322110.
The weak-antichain case is counted by A322138, or A322117 with singletons.
Cf. A002218, A007718, A013922, A030019, A275307, A293994, A304118, A304382, A304887, A305079, A319719, A319721, A321194.

Definition corrected by Gus Wiseman, Feb 05 2021

A322136 Numbers whose number of prime factors counted with multiplicity exceeds half their sum of prime indices by at least 1.

Original entry on oeis.org

4, 8, 12, 16, 24, 32, 36, 40, 48, 64, 72, 80, 96, 108, 112, 120, 128, 144, 160, 192, 216, 224, 240, 256, 288, 320, 324, 336, 352, 360, 384, 400, 432, 448, 480, 512, 576, 640, 648, 672, 704, 720, 768, 800, 832, 864, 896, 960, 972

Offset: 1

Gus Wiseman, Nov 27 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). The sequence lists all Heinz numbers of integer partitions where the number of parts is at least 1 plus half the sum of parts.

Also Heinz numbers of integer partitions that are the vertex-degrees of some hypertree. We allow no singletons in a hypertree, so 2 is not included.

			The sequence of partitions with Heinz numbers in the sequence begins: (11), (111), (211), (1111), (2111), (11111), (2211), (3111), (21111), (111111), (22111), (31111), (211111), (22211), (41111), (32111), (1111111).

Cf. A000569, A025065, A030019, A056156, A056239, A056503, A112798, A181821, A242414, A304382, A320922, A320923, A320924, A320925.

Select[Range[1000],PrimeOmega[#]>=(Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]+2)/2&]

A304383 a(n) = 36*2^n - 5 (n>=1).

Original entry on oeis.org

67, 139, 283, 571, 1147, 2299, 4603, 9211, 18427, 36859, 73723, 147451, 294907, 589819, 1179643, 2359291, 4718587, 9437179, 18874363, 37748731, 75497467, 150994939, 301989883, 603979771, 1207959547, 2415919099, 4831838203, 9663676411, 19327352827, 38654705659, 77309411323

Offset: 1

Emeric Deutsch, May 13 2018

a(n) is the number of edges in the molecular graph NS2[n], defined pictorially in the Ashrafi et al. reference (Fig. 2, where NS2[2] is shown).

Colin Barker, Table of n, a(n) for n = 1..1000
Ali Reza Ashrafi and Parisa Nikzad, Kekulé index and bounds of energy for nanostar dendrimers, Digest J. of Nanomaterials and Biostructures, 4, No. 2, 2009, 383-388.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A304382.

List([1..40],n->36*2^n-5); # Muniru A Asiru, May 13 2018

Maple
```
seq(36*2^n-5, n = 1 .. 40);
```

Vec(x*(67 - 62*x) / ((1 - x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, May 14 2018

From Colin Barker, May 14 2018: (Start)
G.f.: x*(67 - 62*x) / ((1 - x)*(1 - 2*x)).
a(n) = 3*a(n-1) - 2*a(n-2) for n>2.
(End)

A305194 Number of z-forests summing to n. Number of strict integer partitions of n with pairwise indivisible parts and all connected components having clutter density -1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 3, 4, 5, 4, 6, 7, 7, 9, 11, 12, 13, 15, 17, 20, 23, 25, 27, 32, 35, 40, 45, 49, 54, 58, 67, 78, 82, 95, 99, 111, 123, 135, 150, 164, 177, 194, 214, 236, 260, 282, 309, 330

Offset: 1

Gus Wiseman, May 27 2018

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-forest is a strict integer partition with pairwise indivisible parts greater than 1 such that all connected components have clutter density -1.

			The a(17) = 11 z-forests together with the corresponding multiset systems:
       (17): {{7}}
     (15,2): {{2,3},{1}}
     (14,3): {{1,4},{2}}
     (13,4): {{6},{1,1}}
     (12,5): {{1,1,2},{3}}
     (11,6): {{5},{1,2}}
     (10,7): {{1,3},{4}}
      (9,8): {{2,2},{1,1,1}}
   (10,4,3): {{1,3},{1,1},{2}}
    (7,6,4): {{4},{1,2},{1,1}}
  (7,5,3,2): {{4},{3},{2},{1}}

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

Cf. A030019, A048143, A134954, A275307, A285572, A293510, A293993, A293994, A303362, A303837, A303838, A304118, A304382, A305078, A305195.

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];
Table[Length[Select[IntegerPartitions[n],Function[s,UnsameQ@@s&&And@@(Length[#]==1||zreeQ[#]&)/@Table[Select[s,Divisible[m,#]&],{m,zsm[s]}]&&Select[Tuples[s,2],UnsameQ@@#&&Divisible@@#&]=={}]]],{n,50}]

A305195 Number of z-blobs summing to n. Number of connected strict integer partitions of n, with pairwise indivisible parts, that cannot be capped by a z-tree.

Original entry on oeis.org

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, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 2, 1, 1, 1, 3, 2, 2, 2, 1, 1, 3, 3, 3, 1, 1, 1, 4, 5, 6, 2, 1, 1, 4, 6, 7, 2, 2, 6

Offset: 1

Gus Wiseman, May 27 2018

nonn

Caps of a clutter are defined in the link, and the generalization to "multiclutters," where edges can be multisets, is straightforward.

			The a(30) = 2 z-blobs together with the corresponding multiset systems:
     (30): {{1,2,3}}
  (18,12): {{1,2,2},{1,1,2}}
The a(47) = 3 z-blobs together with the corresponding multiset systems:
        (47): {{15}}
  (21,14,12): {{2,4},{1,4},{1,1,2}}
  (20,15,12): {{1,1,3},{2,3},{1,1,2}}
The a(60) = 5 z-blobs together with the corresponding multiset systems:
           (60): {{1,1,2,3}}
        (42,18): {{1,2,4},{1,2,2}}
        (36,24): {{1,1,2,2},{1,1,1,2}}
     (30,18,12): {{1,2,3},{1,2,2},{1,1,2}}
  (21,15,14,10): {{2,4},{2,3},{1,4},{1,3}}
The a(67) = 7 z-blobs together with the corresponding multiset systems:
           (67): {{19}}
     (45,12,10): {{2,2,3},{1,1,2},{1,3}}
     (42,15,10): {{1,2,4},{2,3},{1,3}}
     (40,15,12): {{1,1,1,3},{2,3},{1,1,2}}
     (33,22,12): {{2,5},{1,5},{1,1,2}}
     (28,21,18): {{1,1,4},{2,4},{1,2,2}}
  (24,18,15,10): {{1,1,1,2},{1,2,2},{2,3},{1,3}}

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

Cf. A030019, A048143, A275307, A285572, A293510, A303362, A303837, A303838, A304118, A304382, A304887, A305028, A305078, A305194.

A321271 Number of connected factorizations of n into positive integers > 1 with z-density -1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 01 2018

nonn

These are z-trees (A303837, A305081, A305253, A321279) where we relax the requirement of pairwise indivisibility.
Given a finite multiset S of positive integers greater than 1, let G(S) be the simple labeled graph with vertices the distinct elements of S and with edges between any two vertices that have a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. Then S is said to be connected if G(S) is a connected graph.
The z-density of a factorization S is defined to be Sum_{s in S} (omega(s) - 1) - omega(n), where omega = A001221 and n is the product of S.

			The a(72) = 8 factorizations are (2*2*3*6), (2*2*18), (2*3*12), (2*36), (3*4*6), (3*24), (4*18), (72). Missing from this list but still connected are (2*6*6),(6*12).

Cf. A001055, A001221, A030019, A286518, A303837, A304118, A304382, A305052, A305081, A305193, A305253, A319786, A321229, A321253.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
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[Times@@s];
Table[Length[Select[facs[n],And[zensity[#]==-1,Length[zsm[#]]==1]&]],{n,100}]

A321272 Number of connected multiset partitions with multiset density -1, of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 5, 1, 4, 4, 7, 3, 11, 7, 8, 1, 15, 8, 22, 7, 14, 12, 30, 5, 16, 19, 20, 14, 42, 18, 56, 1, 24, 30, 28, 18, 77, 45, 38, 14

Offset: 1

Gus Wiseman, Nov 01 2018

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.

			Non-isomorphic representatives of the a(2) = 1 through a(15) = 8 multiset partitions:
  {{1}}  {{11}}    {{12}}  {{111}}      {{112}}    {{1111}}
         {{1}{1}}          {{1}{11}}    {{1}{12}}  {{1}{111}}
                           {{1}{1}{1}}             {{11}{11}}
                                                   {{1}{1}{11}}
                                                   {{1}{1}{1}{1}}
.
  {{123}}  {{1122}}      {{1112}}      {{11111}}
           {{1}{122}}    {{1}{112}}    {{1}{1111}}
           {{2}{112}}    {{11}{12}}    {{11}{111}}
           {{1}{2}{12}}  {{1}{1}{12}}  {{1}{1}{111}}
                                       {{1}{11}{11}}
                                       {{1}{1}{1}{11}}
                                       {{1}{1}{1}{1}{1}}
.
  {{1123}}    {{111111}}            {{11112}}        {{11122}}
  {{1}{123}}  {{1}{11111}}          {{1}{1112}}      {{1}{1122}}
  {{12}{13}}  {{11}{1111}}          {{11}{112}}      {{11}{122}}
              {{111}{111}}          {{12}{111}}      {{2}{1112}}
              {{1}{1}{1111}}        {{1}{1}{112}}    {{1}{1}{122}}
              {{1}{11}{111}}        {{1}{11}{12}}    {{1}{2}{112}}
              {{11}{11}{11}}        {{1}{1}{1}{12}}  {{2}{11}{12}}
              {{1}{1}{1}{111}}                       {{1}{1}{2}{12}}
              {{1}{1}{11}{11}}
              {{1}{1}{1}{1}{11}}
              {{1}{1}{1}{1}{1}{1}}

Cf. A007718, A181821, A303837, A304382, A305081, A305936, A318284, A321155, A321229, A321253, A321270, A321271.

a(prime(n)) = A000041(n).

A321279 Number of z-trees with product A181821(n). Number of connected antichains of multisets with multiset density -1, of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 4, 2, 2, 1, 2, 3, 4, 4, 2, 4, 3, 4, 4, 3, 4, 6, 4, 6, 2, 1, 4, 6, 4, 9, 6, 5, 3, 9, 2, 8, 4, 9, 8, 7, 4, 8, 4, 12, 6, 12, 5, 16, 8, 17, 5, 7, 2, 19, 6, 10, 10, 1, 6, 13, 2, 16, 7, 16, 6, 27, 4, 7, 16, 20, 8, 15, 4, 22

Offset: 1

Gus Wiseman, Nov 01 2018

nonn

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.

			The sequence of antichains begins:
   2: {{1}}
   3: {{1,1}}
   3: {{1},{1}}
   4: {{1,2}}
   5: {{1,1,1}}
   5: {{1},{1},{1}}
   6: {{1,1,2}}
   7: {{1,1,1,1}}
   7: {{1,1},{1,1}}
   7: {{1},{1},{1},{1}}
   8: {{1,2,3}}
   9: {{1,1,2,2}}
  10: {{1,1,1,2}}
  10: {{1,1},{1,2}}
  11: {{1,1,1,1,1}}
  11: {{1},{1},{1},{1},{1}}
  12: {{1,1,2,3}}
  12: {{1,2},{1,3}}
  13: {{1,1,1,1,1,1}}
  13: {{1,1,1},{1,1,1}}
  13: {{1,1},{1,1},{1,1}}
  13: {{1},{1},{1},{1},{1},{1}}
  14: {{1,1,1,1,2}}
  14: {{1,2},{1,1,1}}
  15: {{1,1,1,2,2}}
  15: {{1,1},{1,2,2}}
  16: {{1,2,3,4}}

Cf. A001055, A007718, A030019, A181821, A293607, A303837, A304382, A305081, A305936, A318284, A321229, A321270, A321271, A321272.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{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]]]]]]]]];
zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
Table[Length[Select[facs[Times@@Prime/@nrmptn[n]],And[zensity[#]==-1,Length[zsm[#]]==1,Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]=={}]&]],{n,50}]

A303674 Number of connected integer partitions of n > 1 whose distinct parts are pairwise indivisible and whose z-density is -1.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 4, 1, 6, 4, 5, 1, 8, 2, 7, 5, 11, 3, 11, 5, 13, 6, 14, 7, 19, 6, 19, 15, 24, 13, 28, 15, 33, 20, 34, 22, 46, 30, 48, 32, 57, 39, 67, 48, 76, 63, 88, 62, 104, 88, 110, 94, 130, 115, 164, 121, 172, 152, 198, 176, 229, 203, 270, 235, 293, 272, 341, 311, 375, 349, 453, 420, 506, 452, 570, 547

Offset: 1

Gus Wiseman, Jun 04 2018

nonn

The z-density of a multiset S is defined to be Sum_{s in S} (omega(s) - 1) - omega(lcm(S)), where omega = A001221 and lcm is least common multiple.

Given a finite multiset 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 multiset S is said to be connected if G(S) is a connected graph.

			The a(18) = 8 integer partitions are (18), (14,4), (10,8), (9,9), (10,4,4), (6,4,4,4), (3,3,3,3,3,3), (2,2,2,2,2,2,2,2,2).
The a(20) = 7 integer partitions are (20), (14,6), (12,8), (10,6,4), (5,5,5,5), (4,4,4,4,4), (2,2,2,2,2,2,2,2,2,2).

Cf. A030019, A035053, A048143, A134954, A286520, A293510, A293994, A303837, A303838, A304382.

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[IntegerPartitions[n],And[zensity[#]==-1,Length[zsm[#]]==1,Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]=={}]&]],{n,30}]

a(51)-a(81) from Robert Price, Sep 15 2018

cp's OEIS Frontend

A322110 Number of non-isomorphic connected multiset partitions of weight n that cannot be capped by a tree.

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A322118 Number of non-isomorphic connected multiset partitions of weight n with no singletons that cannot be capped by a tree.

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A322136 Numbers whose number of prime factors counted with multiplicity exceeds half their sum of prime indices by at least 1.

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A304383 a(n) = 36*2^n - 5 (n>=1).

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A305194 Number of z-forests summing to n. Number of strict integer partitions of n with pairwise indivisible parts and all connected components having clutter density -1.

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A305195 Number of z-blobs summing to n. Number of connected strict integer partitions of n, with pairwise indivisible parts, that cannot be capped by a z-tree.

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A321271 Number of connected factorizations of n into positive integers > 1 with z-density -1.

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A321272 Number of connected multiset partitions with multiset density -1, of a multiset whose multiplicities are the prime indices of n.

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A321279 Number of z-trees with product A181821(n). Number of connected antichains of multisets with multiset density -1, of a multiset whose multiplicities are the prime indices of n.

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