A304714 -id:A304714 - cp's OEIS frontend

A305193 Number of connected factorizations of n.

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

Offset: 1

Gus Wiseman, May 27 2018

nonn

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

			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).

Antti Karttunen, Table of n, a(n) for n = 1..20736
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A048143, A281116, A286518, A286520, A290103, A303837, A304118, A304714, A304716, A305078, A305079, A319786.

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}]

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

More terms from Antti Karttunen, Nov 07 2018

A322336 Heinz numbers of 2-edge-connected integer partitions.

Original entry on oeis.org

9, 21, 25, 27, 39, 49, 57, 63, 65, 81, 87, 91, 111, 115, 117, 121, 125, 129, 133, 147, 159, 169, 171, 183, 185, 189, 203, 213, 235, 237, 243, 247, 259, 261, 267, 273, 289, 299, 301, 303, 305, 319, 321, 325, 333, 339, 343, 351, 361, 365, 371, 377, 387, 393, 399

Offset: 1

Gus Wiseman, Dec 04 2018

nonn

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

An integer partition is 2-edge-connected if the hypergraph of prime factorizations of its parts is connected and cannot be disconnected by removing any single part. For example (6,6,3,2) is 2-edge-connected but (6,3,2) is not.

			The sequence of all 2-edge-connected integer partitions begins: (2,2), (4,2), (3,3), (2,2,2), (6,2), (4,4), (8,2), (4,2,2), (6,3), (2,2,2,2), (10,2), (6,4), (12,2), (9,3), (6,2,2), (5,5), (3,3,3), (14,2), (8,4), (4,4,2).

Wikipedia, k-edge-connected graph

Cf. A001222, A003963, A013922, A054921, A056239, A095983, A112798, A218970, A275307, A304714, A304716, A305078, A305079, A322335, A322337, A322338.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
twoedQ[sys_]:=And[Length[csm[sys]]==1,And@@Table[Length[csm[Delete[sys,i]]]==1,{i,Length[sys]}]];
Select[Range[100],twoedQ[primeMS/@primeMS[#]]&]

A322387 Number of 2-vertex-connected integer partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 3, 1, 6, 2, 10, 8, 13, 9, 26, 14, 35, 28, 50, 37, 77, 54, 101, 84, 138, 110, 205, 149, 252, 222, 335, 287, 455, 375, 577, 522, 740, 657, 985

Offset: 1

Gus Wiseman, Dec 05 2018

An integer partition is 2-vertex-connected if the prime factorizations of the parts form a connected hypergraph that is still connected if any single prime number is divided out of all the parts (and any parts then equal to 1 are removed).

			The a(14) = 10 2-vertex-connected integer partitions:
  (14)  (8,6)   (6,4,4)   (6,3,3,2)  (6,2,2,2,2)
        (10,4)  (6,6,2)   (6,4,2,2)
        (12,2)  (10,2,2)

Wikipedia, k-vertex-connected graph

Cf. A013922, A095983, A218970, A275307, A304714, A304716, A305078, A305079, A322335, A322336, A322337, A322338, A322388, A322389, A322390.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
vertConn[y_]:=If[Length[csm[primeMS/@y]]!=1,0,Min@@Length/@Select[Subsets[Union@@primeMS/@y],Function[del,Length[csm[DeleteCases[DeleteCases[primeMS/@y,Alternatives@@del,{2}],{}]]]!=1]]];
Table[Length[Select[IntegerPartitions[n],vertConn[#]>1&]],{n,30}]

a(41)-a(42) from Jinyuan Wang, Jun 20 2020

A322390 Number of integer partitions of n with vertex-connectivity 1.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 8, 1, 7, 3, 11, 1, 14, 2, 18, 7, 21, 6, 35, 14, 43, 28, 65, 42, 96, 70, 141, 120, 205, 187, 315, 286, 445, 445, 657

Offset: 1

Gus Wiseman, Dec 05 2018

The vertex-connectivity of an integer partition is the minimum number of primes that must be divided out (and any parts then equal to 1 removed) so that the prime factorizations of the remaining parts form a disconnected (or empty) hypergraph.

			The a(14) = 7 integer partitions are (842), (8222), (77), (4442), (44222), (422222), (2222222).
The a(18) = 14 integer partitions:
  (9,9), (16,2),
  (8,8,2), (10,6,2),
  (8,4,4,2), (9,3,3,3),
  (4,4,4,4,2), (8,4,2,2,2),
  (3,3,3,3,3,3), (4,4,4,2,2,2), (8,2,2,2,2,2),
  (4,4,2,2,2,2,2),
  (4,2,2,2,2,2,2,2),
  (2,2,2,2,2,2,2,2,2).

Wikipedia, k-vertex-connected graph

Cf. A013922, A054921, A095983, A304714, A304716, A305078, A305079, A322335, A322338, A322387, A322389, A322391.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
vertConn[y_]:=If[Length[csm[primeMS/@y]]!=1,0,Min@@Length/@Select[Subsets[Union@@primeMS/@y],Function[del,Length[csm[DeleteCases[DeleteCases[primeMS/@y,Alternatives@@del,{2}],{}]]]!=1]]];
Table[Length[Select[IntegerPartitions[n],vertConn[#]==1&]],{n,20}]

A325105 Number of binary carry-connected subsets of {1...n}.

Original entry on oeis.org

1, 2, 3, 7, 8, 20, 48, 112, 113, 325, 777, 1737, 3709, 7741, 15869, 32253, 32254, 96538, 225798, 485702, 1006338, 2049602, 4137346, 8315266, 16697102, 33465934, 67007886, 134100366, 268301518, 536720590, 1073575118, 2147316942, 2147316943, 6441886323

Offset: 0

Gus Wiseman, Mar 28 2019

nonn

A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion. A subset is binary carry-connected if the graph whose vertices are the elements and whose edges are binary carries is connected.

			The a(0) = 1 through a(4) = 8 subsets:
  {}  {}   {}   {}       {}
      {1}  {1}  {1}      {1}
           {2}  {2}      {2}
                {3}      {3}
                {1,3}    {4}
                {2,3}    {1,3}
                {1,2,3}  {2,3}
                         {1,2,3}

Alois P. Heinz, Table of n, a(n) for n = 0..1023

Cf. A019565, A080572, A247935, A304714, A304716, A305078.
Cf. A325095, A325098, A325099, A325104, A325107, A325118, A325119.
Partial sums of A306299.

h:= proc(n, s) local i, m; m:= n;
      for i in s do m:= Bits[Or](m, i) od; {m}
    end:
g:= (n, s)-> (w-> `if`(w={}, s union {n}, s minus w union
              h(n, w)))(select(x-> Bits[And](n, x)>0, s)):
b:= proc(n, s) option remember; `if`(n=0,
      `if`(nops(s)>1, 0, 1), b(n-1, s)+b(n-1, g(n, s)))
    end:
a:= n-> b(n, {}):
seq(a(n), n=0..35);  # Alois P. Heinz, Mar 31 2019

binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Range[n]],Length[csm[binpos/@#]]<=1&]],{n,0,10}]

a(n) = A306297(n,0) + A306297(n,1). - Alois P. Heinz, Mar 31 2019

a(16)-a(33) from Alois P. Heinz, Mar 31 2019

A304382 Number of z-trees summing to n. Number of connected strict integer partitions of n with pairwise indivisible parts and clutter density -1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 1, 3, 2, 4, 3, 5, 2, 5, 4, 6, 3, 7, 6, 8, 4, 9, 8, 13, 9, 15, 8, 14, 12, 16, 12, 20, 20, 24, 15, 27, 20, 33, 27, 35

Offset: 1

Gus Wiseman, May 21 2018

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 clutter density of a multiset S of positive integers is Sum_{s in S} (omega(s) - 1) - omega(LCM(S)).

			The a(30) = 8 z-trees together with the corresponding multiset systems are the following.
       (30): {{1,2,3}}
     (26,4): {{1,6},{1,1}}
     (22,8): {{1,5},{1,1,1}}
     (21,9): {{2,4},{2,2}}
    (16,14): {{1,1,1,1},{1,4}}
   (15,9,6): {{2,3},{2,2},{1,2}}
  (14,10,6): {{1,4},{1,3},{1,2}}
  (12,10,8): {{1,1,2},{1,3},{1,1,1}}

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

Cf. A000009, A006126, A048143, A054921, A112798, A285572, A286518, A286520, A293993, A303362, A303837, A304714, A304716, A305194, 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];
strConnAnti[n_]:=Select[IntegerPartitions[n],UnsameQ@@#&&Length[zsm[#]]==1&&Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]=={}&];
Table[Length[Select[strConnAnti[n],Length[#]==1||zreeQ[#]&]],{n,20}]

A325119 Heinz numbers of binary carry-connected strict integer partitions.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 11, 13, 15, 17, 19, 22, 23, 29, 30, 31, 34, 37, 39, 41, 43, 46, 47, 51, 53, 55, 59, 61, 62, 65, 67, 71, 73, 77, 79, 82, 83, 85, 87, 89, 91, 93, 94, 97, 101, 102, 103, 107, 109, 110, 113, 115, 118, 119, 127, 129, 130, 131, 134, 137, 139, 141

Offset: 1

Gus Wiseman, Mar 28 2019

nonn

A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion. An integer partition is binary carry-connected if the graph whose vertices are the parts and whose edges are binary carries is connected.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are squarefree numbers whose prime indices are binary carry-connected. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   7: {4}
  10: {1,3}
  11: {5}
  13: {6}
  15: {2,3}
  17: {7}
  19: {8}
  22: {1,5}
  23: {9}
  29: {10}
  30: {1,2,3}
  31: {11}
  34: {1,7}
  37: {12}
  39: {2,6}
  41: {13}
  43: {14}

Cf. A019565, A048143, A056239, A112798, A304714, A304716, A305078.

Cf. A325098, A325099, A325100, A325105, A325110, A325118.

binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Select[Range[100],SquareFreeQ[#]&&Length[csm[binpos/@PrimePi/@First/@FactorInteger[#]]]<=1&]

A322337 Number of strict 2-edge-connected integer partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 4, 0, 4, 3, 5, 0, 9, 0, 10, 5, 11, 1, 18, 3, 17, 8, 22, 3, 35, 5, 32, 17, 39, 16, 59, 14, 58, 33, 75, 28, 103, 35, 106, 71, 125, 63, 174, 81, 192, 127, 220, 130, 294, 170, 325, 237, 378, 257, 504

Offset: 1

Gus Wiseman, Dec 04 2018

nonn

An integer partition is 2-edge-connected if the hypergraph of prime factorizations of its parts is connected and cannot be disconnected by removing any single part.

			The a(24) = 18 strict 2-edge-connected integer partitions of 24:
  (15,9)   (10,8,6)   (10,8,4,2)
  (16,8)   (12,8,4)   (12,6,4,2)
  (18,6)   (12,9,3)
  (20,4)   (14,6,4)
  (21,3)   (14,8,2)
  (22,2)   (15,6,3)
  (14,10)  (16,6,2)
           (18,4,2)
           (12,10,2)

Wikipedia, k-edge-connected graph

Cf. A007718, A013922, A054921, A095983, A218970, A275307, A286518, A304714, A304716, A305078, A305079, A322335, A322336.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
twoedQ[sys_]:=And[Length[csm[sys]]==1,And@@Table[Length[csm[Delete[sys,i]]]==1,{i,Length[sys]}]];
Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,twoedQ[primeMS/@#]]&]],{n,30}]

A329552 Smallest MM-number of a connected set of n sets.

Original entry on oeis.org

1, 2, 39, 195, 5655, 62205, 2674815

Offset: 0

Gus Wiseman, Nov 17 2019

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The sequence of terms together with their corresponding systems begins:
        1: {}
        2: {{}}
       39: {{1},{1,2}}
      195: {{1},{2},{1,2}}
     5655: {{1},{2},{1,2},{1,3}}
    62205: {{1},{2},{3},{1,2},{1,3}}
  2674815: {{1},{2},{3},{1,2},{1,3},{1,4}}

MM-numbers of connected set-systems are A328514.
The weight of the system with MM-number n is A302242(n).
Connected numbers are A305078.
Maximum connected divisor is A327076.
BII-numbers of connected sets of sets are A326749.
The smallest BII-number of a connected set of n sets is A329625(n).
Allowing edges to have repeated vertices gives A329553.
Requiring the edges to form an antichain gives A329555.
The smallest MM-number of a set of n nonempty sets is A329557(n).
Cf. A048143, A056239, A112798, A302494, A304714, A304716, A305079, A322389, A328513, A329554, A329556, A329558.
Classes of MM-numbers: A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A329559 (clutters).

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]]]]]]]]];
da=Select[Range[10000],SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&Length[zsm[primeMS[#]]]<=1&];
Table[da[[Position[PrimeOmega/@da,n][[1,1]]]],{n,First[Split[Union[PrimeOmega/@da],#2==#1+1&]]}]

A322306 Number of connected divisors of n. Number of connected submultisets of the n-th multiset multisystem (A302242).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 03 2018

nonn

A prime index of n is a number m such that prime(m) divides n. A positive integer is connected if its prime indices are connected (see A305078).

			The a(1365) = 12 divisors are 3, 5, 7, 13, 21, 39, 65, 91, 195, 273, 455, 1365. These correspond to the following connected submultisets of {{1},{2},{1,1},{1,2}}.
     3: {{1}}
     5: {{2}}
     7: {{1,1}}
    13: {{1,2}}
    21: {{1},{1,1}}
    39: {{1},{1,2}}
    65: {{2},{1,2}}
    91: {{1,1},{1,2}}
   195: {{1},{2},{1,2}}
   273: {{1},{1,1},{1,2}}
   455: {{2},{1,1},{1,2}}
  1365: {{1},{2},{1,1},{1,2}}

Cf. A003963, A054921, A112798, A286518, A290103, A301957, A302242, A304714, A304716, A305078, A316556, A322307.

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]]]]]]]]];
Table[Length[Select[Union[Subsets[primeMS[n]]],Length[zsm[#]]==1&]],{n,50}]

cp's OEIS Frontend

A305193 Number of connected factorizations of n.

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A322336 Heinz numbers of 2-edge-connected integer partitions.

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A322387 Number of 2-vertex-connected integer partitions of n.

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A322390 Number of integer partitions of n with vertex-connectivity 1.

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A325105 Number of binary carry-connected subsets of {1...n}.

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A304382 Number of z-trees summing to n. Number of connected strict integer partitions of n with pairwise indivisible parts and clutter density -1.

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A325119 Heinz numbers of binary carry-connected strict integer partitions.

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A322337 Number of strict 2-edge-connected integer partitions of n.

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