A305078 -id:A305078 - cp's OEIS frontend

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

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}]

A320462 MM-numbers of labeled multigraphs with loops spanning an initial interval of positive integers.

Original entry on oeis.org

1, 7, 13, 49, 91, 161, 169, 299, 329, 343, 377, 611, 637, 667, 1127, 1183, 1261, 1363, 1937, 2021, 2093, 2117, 2197, 2303, 2401, 2639, 3703, 3887, 4277, 4459, 4669, 4901, 6877, 7567, 7889, 7943, 8281, 8671, 8827, 9541, 10933, 13559, 14053, 14147, 14651, 14819

Offset: 1

Gus Wiseman, Oct 13 2018

nonn

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 multisystem 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 multisystem with MM-number 78 is {{},{1},{1,2}}.

			The sequence of terms together with their multiset multisystems begins:
     1: {}
     7: {{1,1}}
    13: {{1,2}}
    49: {{1,1},{1,1}}
    91: {{1,1},{1,2}}
   161: {{1,1},{2,2}}
   169: {{1,2},{1,2}}
   299: {{2,2},{1,2}}
   329: {{1,1},{2,3}}
   343: {{1,1},{1,1},{1,1}}
   377: {{1,2},{1,3}}
   611: {{1,2},{2,3}}
   637: {{1,1},{1,1},{1,2}}
   667: {{2,2},{1,3}}
  1127: {{1,1},{1,1},{2,2}}
  1183: {{1,1},{1,2},{1,2}}

Eric Weisstein's World of Mathematics, Simple Graph

Cf. A003963, A055932, A056239, A112798, A255906, A290103, A302242, A305052, A305078.

Cf. A320456, A320458, A320459, A320461, A320533.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
Select[Range[10000],And[normQ[primeMS/@primeMS[#]],And@@(Length[primeMS[#]]==2&/@primeMS[#])]&]

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

A328336 Numbers with no consecutive prime indices relatively prime.

Original entry on oeis.org

1, 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, Oct 14 2019

nonn

First differs from A318978 in having 897, with prime indices {2, 6, 9}.
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 Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of partitions no consecutive parts relatively prime (A328187).
Besides the initial 1 this differs from A305078: 47541=897*prime(16) is in A305078 but not in this set. - Andrey Zabolotskiy, Nov 13 2019

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   7: {4}
   9: {2,2}
  11: {5}
  13: {6}
  17: {7}
  19: {8}
  21: {2,4}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  31: {11}
  37: {12}
  39: {2,6}
  41: {13}
  43: {14}

Numbers with consecutive prime indices relatively prime are A328335.
Strict partitions with no consecutive parts relatively prime are A328220.
Numbers with relatively prime prime indices are A289509.
Cf. A000837, A056239, A078374, A112798, A281116, A289508, A318981, A328168, A328169, A328172, A328187, A328188.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!MatchQ[primeMS[#],{_,x_,y_,_}/;GCD[x,y]==1]&]

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&]

A317077 Number of connected multiset partitions of normal multisets of size n.

Original entry on oeis.org

1, 1, 3, 8, 28, 110, 519, 2749, 16317, 106425, 755425, 5781956, 47384170, 413331955, 3818838624, 37213866876, 381108145231, 4088785729738, 45829237977692, 535340785268513, 6502943193997922, 81984445333355812, 1070848034863526547, 14467833457108560375, 201894571410270034773

Offset: 0

Gus Wiseman, Jul 20 2018

nonn

A multiset is normal if it spans an initial interval of positive integers.

			The a(3) = 8 connected multiset partitions are (111), (1)(11), (1)(1)(1), (122), (2)(12), (112), (1)(12), (123).

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A007716, A007718, A048143, A293994, A303837, A303838, A304716, A305078.

Cf. A317073, A317075, A317078, A317079, A317080.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],multijoin@@s[[c[[1]]]]]]]]];
allnorm[n_]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Length/@Table[Join@@Table[Select[mps[m],Length[csm[#]]==1&],{m,allnorm[n]}],{n,8}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
Connected(v)={my(u=vector(#v));for(n=1, #u, u[n]=v[n] - sum(k=1, n-1, binomial(n-1,k)*v[k]*u[n-k]));u}
seq(n)={my(u=vector(n, k, x*Ser(EulerT(vector(n,i,binomial(i+k-1,i)))))); Vec(1+vecsum(Connected(vector(n, k, sum(i=1, k, (-1)^(k-i)*binomial(k,i)*u[i])))))} \\ Andrew Howroyd, Jan 16 2023

Terms a(9) and beyond from Andrew Howroyd, Jan 16 2023

A329557 Smallest MM-number of a set of n nonempty sets.

Original entry on oeis.org

1, 3, 15, 165, 2145, 36465, 1057485, 32782035, 1344063435, 57794727705, 2716352202135, 160264779925965, 10737740255039655, 783855038617894815, 61924548050813690385, 5139737488217536301955, 519113486309971166497455, 56583370007786857148222595, 6393920810879914857749153235

Offset: 0

Gus Wiseman, Nov 17 2019

nonn

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: {}
        3: {{1}}
       15: {{1},{2}}
      165: {{1},{2},{3}}
     2145: {{1},{2},{3},{1,2}}
    36465: {{1},{2},{3},{1,2},{4}}
  1057485: {{1},{2},{3},{1,2},{4},{1,3}}

MM-numbers of sets of sets are A302494.
MM-numbers of sets of nonempty sets are A329629.
The version allowing empty sets is A329558.
The version without singletons is A329554.
Cf. A056239, A072639, A112798, A302242, A326031, A329552, A329555, A329556.
Other 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}]]]];
dae=Select[Range[10000],SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&FreeQ[primeMS[#],1]&];
Table[dae[[Position[PrimeOmega/@dae,k][[1,1]]]],{k,First[Split[Union[PrimeOmega/@dae],#2==#1+1&]]}]

a(n) = my(k=1); prod(i=1, n, until(issquarefree(k), k++); prime(k)); \\ Jinyuan Wang, Feb 23 2025

a(n) = A329558(n + 1)/2.

More terms from Jinyuan Wang, Feb 23 2025

A371291 Numbers whose binary indices are connected, where two numbers are connected iff they have a common factor.

Original entry on oeis.org

1, 2, 4, 8, 10, 16, 32, 34, 36, 38, 40, 42, 44, 46, 64, 128, 130, 136, 138, 160, 162, 164, 166, 168, 170, 172, 174, 256, 260, 288, 290, 292, 294, 296, 298, 300, 302, 416, 418, 420, 422, 424, 426, 428, 430, 512, 514, 520, 522, 528, 530, 536, 538, 544, 546, 548

Offset: 1

Gus Wiseman, Mar 27 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

The empty set is not considered connected.

			The terms together with their binary expansions and binary indices begin:
    1:          1 ~ {1}
    2:         10 ~ {2}
    4:        100 ~ {3}
    8:       1000 ~ {4}
   10:       1010 ~ {2,4}
   16:      10000 ~ {5}
   32:     100000 ~ {6}
   34:     100010 ~ {2,6}
   36:     100100 ~ {3,6}
   38:     100110 ~ {2,3,6}
   40:     101000 ~ {4,6}
   42:     101010 ~ {2,4,6}
   44:     101100 ~ {3,4,6}
   46:     101110 ~ {2,3,4,6}
   64:    1000000 ~ {7}
  128:   10000000 ~ {8}
  130:   10000010 ~ {2,8}
  136:   10001000 ~ {4,8}
  138:   10001010 ~ {2,4,8}
  160:   10100000 ~ {6,8}
  162:   10100010 ~ {2,6,8}
  164:   10100100 ~ {3,6,8}

For prime indices of each prime index we have A305078.
The opposite version is A325118.
For binary indices of each binary index we have A326749.
The pairwise indivisible case is A371294, opposite A371445.
Positions of ones in A371452.
A001187 counts connected graphs.
A007718 counts non-isomorphic connected multiset partitions.
A048143 counts connected antichains of sets.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A087086 lists numbers whose binary indices are pairwise indivisible.
A096111 gives product of binary indices.
A326964 counts connected set-systems, covering A323818.
Cf. A000040, A000720, A001222, A305079, A326753, A371446.

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,1000],Length[csm[prix/@bpe[#]]]==1&]

cp's OEIS Frontend

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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A320462 MM-numbers of labeled multigraphs with loops spanning an initial interval of positive integers.

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

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A328336 Numbers with no consecutive prime indices relatively prime.

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

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Mathematica

A317077 Number of connected multiset partitions of normal multisets of size n.

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