A327076 -id:A327076 - cp's OEIS frontend

A327147 Smallest BII-number of a set-system with spanning edge-connectivity n.

0, 1, 52, 116, 3952, 8052

Offset: 0

Gus Wiseman, Sep 01 2019

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. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system (finite set of finite nonempty sets) has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

The spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a set-system that is disconnected or covers fewer vertices.

			The sequence of terms together with their corresponding set-systems begins:
     0: {}
     1: {{1}}
    52: {{1,2},{1,3},{2,3}}
   116: {{1,2},{1,3},{2,3},{1,2,3}}
  3952: {{1,3},{2,3},{1,4},{2,4},{3,4},{1,2,3},{1,2,4}}
  8052: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,2,3},{1,2,4},{1,3,4}}

The same for cut-connectivity is A327234.
The same for non-spanning edge-connectivity is A002450.
The spanning edge-connectivity of the set-system with BII-number n is A327144(n).
Cf. A000120, A048793, A070939, A323818, A326031, A326786, A326787, A327041, A327069, A327076, A327108, A327111, A327130, A327145.

A327398 Maximum connected squarefree divisor of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 20 2019

nonn

A squarefree number with prime factorization prime(m_1) * ... * prime(m_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.

			The connected squarefree divisors of 189 are {1, 3, 7, 21}, so a(189) = 21.

Gus Wiseman, Sequences counting and encoding certain classes of multisets

The maximum connected divisor of n is A327076(n).
The maximum squarefree divisor of n is A007947(n).
Connected numbers are A305078.
Connected squarefree numbers are A328513.
Cf. A000005, A001221, A302242, A302494, A304714, A305079, A327656, A328514.

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[Max[Select[Divisors[n],SquareFreeQ[#]&&Length[zsm[primeMS[#]]]<=1&]],{n,100}]

A329553 Smallest MM-number of a connected set of n multisets.

Original entry on oeis.org

1, 2, 21, 195, 1365, 25935, 435435

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: {{}}
      21: {{1},{1,1}}
     195: {{1},{2},{1,2}}
    1365: {{1},{2},{1,1},{1,2}}
   25935: {{1},{2},{1,1},{1,2},{1,1,1}}
  435435: {{1},{2},{1,1},{3},{1,2},{1,3}}

MM-numbers of connected sets of sets 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 set-systems are A326749.
The smallest BII-number of a connected set-system is A329625.
The case of strict edges is A329552.
The smallest MM-number of a set of n nonempty sets is A329557(n).
Cf. A056239, A112798, A302494, A305079, A329554, A329555, 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[Subsets[Range[Length[s]],{2}],GCD@@s[[#]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
dae=Select[Range[100000],SquareFreeQ[#]&&Length[zsm[primeMS[#]]]<=1&];
Table[dae[[Position[PrimeOmega/@dae,k][[1,1]]]],{k,First[Split[Union[PrimeOmega/@dae],#2==#1+1&]]}]

A327392 Irregular triangle read by rows giving the connected components of the prime indices of n.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 2, 4, 1, 1, 1, 2, 1, 3, 5, 1, 1, 2, 6, 1, 4, 2, 3, 1, 1, 1, 1, 7, 1, 2, 8, 1, 1, 3, 4, 1, 5, 9, 1, 1, 1, 2, 3, 1, 6, 2, 1, 1, 4, 10, 1, 2, 3, 11, 1, 1, 1, 1, 1, 2, 5, 1, 7, 3, 4, 1, 1, 2, 12, 1, 8, 6, 1, 1, 1, 3, 13, 1, 4, 14, 1, 1, 5, 2, 3

Offset: 1

Gus Wiseman, Oct 03 2019

First differs from A112798 at a(13) = 1, A112798(13) = 2.
The terms of each row are pairwise coprime.
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.
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.

			Triangle begins:
  {}
  1
  2
  1 1
  3
  1 2
  4
  1 1 1
  2
  1 3
  5
  1 1 2
  6
  1 4
  2 3
  1 1 1 1
  7
  1 2
  8
  1 1 3
  4
  1 5
  9
  1 1 1 2
  3
  1 6
  2
  1 1 4

Row lengths are A305079.

Cf. A000005, A056239, A112798, A218970, A304716, A302242, A305078, A327076.

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[zsm[primeMS[n]],{n,30}]

A327395 Quotient of n over the maximum connected divisor of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 3, 8, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 6, 1, 16, 3, 2, 5, 4, 1, 2, 1, 8, 1, 2, 1, 4, 5, 2, 1, 16, 1, 2, 3, 4, 1, 2, 5, 8, 1, 2, 1, 12, 1, 2, 1, 32, 1, 6, 1, 4, 3, 10, 1, 8, 1, 2, 3, 4, 7, 2, 1, 16, 1, 2, 1, 4, 5

Offset: 1

Gus Wiseman, Sep 15 2019

nonn

Requires A305079(n) steps to reach 1, the only fixed point.

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.

Gus Wiseman, Sequences counting and encoding certain classes of multisets

See link for additional crossrefs.
Positions of 1's are A305078.
Positions of 2's are 2 * A305078.
Cf. A000005, A304716, A304714, A305079, A327076.

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]]]]]]]]];
maxcon[n_]:=Max[Select[Divisors[n],Length[zsm[primeMS[#]]]<=1&]];
Table[n/maxcon[n],{n,100}]

a(n) = n/A327076(n).

A328512 Number of distinct connected components of the multiset of multisets with MM-number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 20 2019

nonn

For n > 1, the first appearance of n is 2 * A080696(n - 1).

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 multiset of multisets with MM-number 1508 is {{},{},{1,2},{1,3}}, which has the 3 connected components {{}}, {{}}, and {{1,2},{1,3}}, two of which are distinct, so a(1508) = 2.
The multiset of multisets with MM-number 12818 is {{},{1,2},{4},{1,3}}, which has the 3 connected components {{}}, {{1,2},{1,3}}, and {{4}}, so a(12818) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to prime indices in the factorization of n.

Positions of 0's and 1's are A305078 together with all powers of 2.
Connected numbers are A305078.
Connected components are A305079.
Cf. A007814, A056239, A112798, A286518, A302242, A304714, A304716, A322389, A327076, A328513.

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[Union[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));
A328512(n) = if(!(n%2), 1+A328512(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, Jan 28 2025

If n is even, a(n) = A305079(n) - A007814(n) + 1; otherwise, a(n) = A305079(n).

Data section extended to a(105) by Antti Karttunen, Jan 28 2025

A327390 Number of connected divisors of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 15 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. The maximum connected divisor of n is A327076(n).

Gus Wiseman, Sequences counting and encoding certain classes of multisets

See link for additional cross-references.

Cf. A000005, A001221, A033273, A286518, A304714, A304716.

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[Select[Divisors[n],Length[zsm[primeMS[#]]]<=1&]],{n,100}]

A327519 Number of factorizations of A305078(n - 1), the n-th connected number, into connected numbers > 1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 21 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.

			The a(190) = 8 factorizations of 585 together with the corresponding multiset partitions of {2,2,3,6}:
  (3*3*5*13)  {{2},{2},{3},{6}}
  (3*3*65)    {{2},{2},{3,6}}
  (3*5*39)    {{2},{3},{2,6}}
  (3*195)     {{2},{2,3,6}}
  (5*9*13)    {{3},{2,2},{6}}
  (5*117)     {{3},{2,2,6}}
  (9*65)      {{2,2},{3,6}}
  (585)       {{2,2,3,6}}

Gus Wiseman, Sequences counting and encoding certain classes of multisets

See link for additional cross-references.

Cf. A286518, A302569, A304714, A304716, A305078, A305079, A327076.

nn=100;
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]]]]]]]]];
facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
y=Select[Range[nn],Length[zsm[primeMS[#]]]==1&];
Table[Length[facsusing[y,n]],{n,y}]

cp's OEIS Frontend

A327147 Smallest BII-number of a set-system with spanning edge-connectivity n.

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A327398 Maximum connected squarefree divisor of n.

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A329553 Smallest MM-number of a connected set of n multisets.

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A327392 Irregular triangle read by rows giving the connected components of the prime indices of n.

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A327395 Quotient of n over the maximum connected divisor of n.

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A328512 Number of distinct connected components of the multiset of multisets with MM-number n.

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A327390 Number of connected divisors of n.

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A327519 Number of factorizations of A305078(n - 1), the n-th connected number, into connected numbers > 1.

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