A001187 -id:A001187 - cp's OEIS frontend

A327200 Number of labeled graphs with n vertices and non-spanning edge-connectivity >= 2.

0, 0, 0, 4, 42, 718, 26262, 1878422, 256204460, 67525498676, 34969833809892, 35954978661632864, 73737437034063350534, 302166248212488958298674, 2475711390267267917290354410, 40563960064630744031043287569378, 1329219366981359393514586291328267704

Offset: 0

Gus Wiseman, Sep 01 2019

nonn

The non-spanning edge-connectivity of a graph is the minimum number of edges that must be removed to obtain a graph whose edge-set is disconnected or empty.

Gus Wiseman, The a(4) = 42 graphs with non-spanning edge-connectivity >= 2.

Row sums of A327148 if the first two columns are removed.
BII-numbers of set-systems with non-spanning edge-connectivity >= 2 are A327102.
Graphs with non-spanning edge-connectivity 1 are A327231.
Cf. A001187, A006129, A095983, A182100, A322395, A326787, A327076, A327079, A327097, A327099, A327236.

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]]]]]]]]];
eConn[sys_]:=If[Length[csm[sys]]!=1,0,Length[sys]-Max@@Length/@Select[Union[Subsets[sys]],Length[csm[#]]!=1&]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],eConn[#]>=2&]],{n,0,5}]

Binomial transform of A322395, if we assume A322395(0) = A322395(1) = A322395(2) = 0.

A370317 Number of labeled graphs with n vertices (allowing isolated vertices) and n edges, such that the edge set is connected.

Original entry on oeis.org

1, 0, 0, 1, 15, 252, 4905, 110715, 2864148, 83838720, 2744568522, 99463408335, 3955626143040, 171344363805582, 8031863998136355, 405150528051451000, 21884686370917378050, 1260420510502767861840, 77105349570138633021624, 4993117552678619556356085

Offset: 0

Gus Wiseman, Feb 17 2024

nonn

			The a(0) = 0 through a(4) = 15 graphs:
  {}  .  .  {{1,2},{1,3},{2,3}}  {{1,2},{1,3},{1,4},{2,3}}
                                 {{1,2},{1,3},{1,4},{2,4}}
                                 {{1,2},{1,3},{1,4},{3,4}}
                                 {{1,2},{1,3},{2,3},{2,4}}
                                 {{1,2},{1,3},{2,3},{3,4}}
                                 {{1,2},{1,3},{2,4},{3,4}}
                                 {{1,2},{1,4},{2,3},{2,4}}
                                 {{1,2},{1,4},{2,3},{3,4}}
                                 {{1,2},{1,4},{2,4},{3,4}}
                                 {{1,2},{2,3},{2,4},{3,4}}
                                 {{1,3},{1,4},{2,3},{2,4}}
                                 {{1,3},{1,4},{2,3},{3,4}}
                                 {{1,3},{1,4},{2,4},{3,4}}
                                 {{1,3},{2,3},{2,4},{3,4}}
                                 {{1,4},{2,3},{2,4},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..100

The covering case is A057500.
This is the connected case of A116508.
Allowing any number of edges gives A287689.
Counting only covered vertices gives A370318.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, connected A001187.
A369192 counts graphs with at most n edges, covering A369191.
Cf. A000169, A001862, A054780, A062740, A129271, A367863, A369193, A369197.

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]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]],Length[#]==n&&Length[csm[#]]<=1&]], {n,0,5}]

a(n)=n!*polcoef(polcoef(exp(x + O(x*x^n))*(1 + log(sum(k=0, n, (1 + y + O(y*y^n))^binomial(k,2)*x^k/k!, O(x*x^n)))), n), n) \\ Andrew Howroyd, Feb 19 2024

a(n) = n!*[x^n][y^n] exp(x)*(1 + log(Sum_{k>=0} (1 + y)^binomial(k, 2)*x^k/k!)). - Andrew Howroyd, Feb 19 2024

a(8) onwards from Andrew Howroyd, Feb 19 2024

A002032 Number of n-colored connected graphs on n labeled nodes.

Original entry on oeis.org

1, 1, 2, 24, 912, 87360, 19226880, 9405930240, 10142439229440, 24057598104207360, 125180857812868300800, 1422700916050060841779200, 35136968950395142864227532800, 1876028272361273394915958613606400, 215474119792145796020405035320528076800

Offset: 0

N. J. A. Sloane

nonn

Every connected graph on n nodes can be colored with n colors in exactly n! ways, so this sequence is just n! * A001187(n). - Andrew Howroyd, Dec 03 2018

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 0..50
R. C. Read, E. M. Wright, Colored graphs: A correction and extension, Canad. J. Math. 22 1970 594-596.

Cf. A002027. A002028, A002029, A002030, A002031.

Cf. A001187, A322278, A322279, A322280.

(* b = A001187 *) b[n_] := b[n] = If[n == 0, 1, 2^(n(n-1)/2) - Sum[k* Binomial[n, k]*2^((n-k)(n-k-1)/2)*b[k], {k, 1, n-1}]/n];
a[n_] := n! b[n];
Array[a, 14] (* Jean-François Alcover, Aug 16 2019, using Alois P. Heinz's code for A001187 *)

seq(n) = {Vec(serlaplace(serlaplace(1 + log(sum(k=0, n, 2^binomial(k, 2)*x^k/k!, O(x*x^n))))))} \\ Andrew Howroyd, Dec 03 2018

a(n) = n!*A001187(n). - Andrew Howroyd, Dec 03 2018
Define M_0(k)=1, M_n(0)=0, M_n(k) = Sum_{r=0..n} C(n,r)*2^(r*(n-r))*M_r(k-1) [M_n(k) = A322280(n,k)], m_n(k) = M_n(k) -Sum_{r=1..n-1} C(n-1,r-1)*m_r(k)*M_{n-r}(k) [m_n(k) = A322279(n,k)], f_n(k) = Sum_{r=1..k} (-1)^(k-r)*C(k,r)*m_n(r). This sequence gives a(n) = f_n(n). - Sean A. Irvine, May 29 2013, edited Andrew Howroyd, Dec 03 2018
The above formula is referenced by sequences A002027-A002030, A002031. - Andrew Howroyd, Dec 03 2018

More terms from Sean A. Irvine, May 29 2013
Name clarified by Andrew Howroyd, Dec 03 2018
a(0)=1 prepended by Andrew Howroyd, Jan 05 2024

A123527 Triangular array T(n,k) giving number of connected graphs with n labeled nodes and k edges (n >= 1, n-1 <= k <= n(n-1)/2).

Original entry on oeis.org

1, 1, 3, 1, 16, 15, 6, 1, 125, 222, 205, 120, 45, 10, 1, 1296, 3660, 5700, 6165, 4945, 2997, 1365, 455, 105, 15, 1, 16807, 68295, 156555, 258125, 331506, 343140, 290745, 202755, 116175, 54257, 20349, 5985, 1330, 210, 21, 1, 262144, 1436568

Offset: 1

N. J. A. Sloane, Nov 13 2006

			Triangle begins:
n = 1
  k = 0: 1
  ****** total(1) = 1
n = 2
  k = 1: 1
  ****** total(2) = 1
n = 3
  k = 2: 3
  k = 3: 1
  ****** total(3) = 4
n = 4
  k = 3: 16
  k = 4: 15
  k = 5:  6
  k = 6:  1
  ****** total(4) = 38
n = 5
  k = 4: 125
  k = 5: 222
  k = 6: 205
  k = 7: 120
  k = 8:  45
  k = 9:  10
  k = 10:  1
  ****** total(5) = 728

Cowan, D. D.; Mullin, R. C.; Stanton, R. G. Counting algorithms for connected labelled graphs. Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1975), pp. 225-236. Congressus Numerantium, No. XIV, Utilitas Math., Winnipeg, Man., 1975. MR0414417 (54 #2519). - From N. J. A. Sloane, Apr 06 2012
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1977.

Seiichi Manyama, Rows n = 1..40, flattened (rows 1..14 from R. W. Robinson, rows 15..20 from Henrique G. G. Pereira)
T. Yanagita, T. Ichinomiya, Thermodynamic Characterization of Synchronization-Optimized Oscillator-Networks, arXiv preprint arXiv:1409.1979 [nlin.AO], 2014.

See A062734 for another version with more information. Row sums give A001187.

nn = 8; a = Sum[(1 + y)^Binomial[n, 2] x^n/n!, {n, 0, nn}]; f[list_] := Select[list, # > 0 &]; Flatten[Map[f, Drop[Range[0, nn]! CoefficientList[Series[Log[a], {x, 0, nn}], {x, y}],1]]] (* Geoffrey Critzer, Dec 08 2011 *)
T[ n_, k_] := If[ n < 1, 0, Coefficient[ n! SeriesCoefficient[ Log[ Sum[ (1 + y)^Binomial[m, 2] x^m/m!, {m, 0, n}]], {x, 0, n}], y, n - 1 + k]]; (* Michael Somos, Aug 12 2017 *)

For k >= C(n-1, 2) + 1 (not smaller!), T(n,k) = C(C(n,2),k) where C(n,k) is the binomial coefficient. See A084546. - Geoffrey Critzer, Dec 08 2011

A129584 Number of unlabeled bi-point-determining graphs: graphs in which no two vertices have the same neighborhoods or the same augmented neighborhoods (the augmented neighborhood of a vertex is the neighborhood of the vertex union the vertex itself).

Original entry on oeis.org

1, 0, 0, 1, 6, 36, 324, 5280, 156088, 8415760

Offset: 1

Ji Li (vieplivee(AT)hotmail.com), May 07 2007

This is the unlabeled case of bi-point-determining graphs, which are basically graphs that are both point-determining (no two vertices have the same neighborhoods) and co-point-determining (graphs whose complements are point-determining)

Ira M. Gessel and Ji Li, Enumeration of point-determining Graphs, Journal of Combinatorial Theory, Series A 118 (2011) 591-612.

Cf. graphs: labeled A006125, unlabeled A000568; connected graphs: labeled A001187, unlabeled A001349; point-determining graphs: labeled A006024, unlabeled A004110; connected point-determining graphs: labeled A092430, unlabeled A004108; connected co-point-determining graphs: labeled A079306, unlabeled A004108; bi-point-determining graphs: labeled A129583, unlabeled A129584; connected bi-point-determining graphs: labeled A129585, unlabeled A129586; phylogenetic trees: labeled A000311, unlabeled A000669.

A324464 Number of connected graphical necklaces with n vertices.

Original entry on oeis.org

1, 0, 1, 2, 13, 148, 4530, 266614, 31451264, 7366255436, 3449652145180, 3240150686268514, 6112883022923529310, 23174784819204929919428, 176546343645071836902594288, 2701845395848698682311893154024, 83036184895986451215378727412638816, 5122922885438069578928905234650082410736

Offset: 0

Gus Wiseman, Mar 01 2019

nonn

A graphical necklace is a simple graph that is minimal among all n rotations of the vertices. Alternatively, it is an equivalence class of simple graphs under rotation of the vertices. These are a kind of partially labeled graphs.

			Inequivalent representatives of the a(2) = 1 through a(4) = 13 graphical necklaces:
  {{12}}  {{12}{13}}      {{12}{13}{14}}
          {{12}{13}{23}}  {{12}{13}{24}}
                          {{12}{13}{34}}
                          {{12}{14}{23}}
                          {{12}{24}{34}}
                          {{12}{13}{14}{23}}
                          {{12}{13}{14}{24}}
                          {{12}{13}{14}{34}}
                          {{12}{13}{24}{34}}
                          {{12}{14}{23}{34}}
                          {{12}{13}{14}{23}{24}}
                          {{12}{13}{14}{23}{34}}
                          {{12}{13}{14}{23}{24}{34}}

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(4) = 13 connected graphical necklaces.
Gus Wiseman, The a(5) = 148 connected graphical necklaces.

Cf. A000031, A000939, A001187, A006125, A006129, A008965, A184271, A192332, A275527, A323858, A323859, A323870, A324461, A324462, A324463.

rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+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[Subsets[Range[n],{2}]],And[Union@@#==Range[n],Length[csm[#]]<=1,#=={}||#==First[Sort[Table[Nest[rotgra[#,n]&,#,j],{j,n}]]]]&]],{n,0,5}]

\\ B(n,d) is graphs on n*d points invariant under 1/d rotation.
B(n,d)={2^(n*(n-1)*d/2 + n*(d\2))}
D(n,d)={my(v=vector(n, i, B(i,d)), u=vector(n)); for(n=1, #u, u[n]=v[n] - sum(k=1, n-1, binomial(n-1, k)*v[k]*u[n-k])); sumdiv(n, e, eulerphi(d*e) * moebius(e) * u[n/e] * e^(n/e-1))}
a(n)={if(n<=1, n==0, sumdiv(n, d, D(n/d,d))/n)} \\ Andrew Howroyd, Jan 24 2023

Terms a(7) and beyond from Andrew Howroyd, Jan 24 2023

A327362 Number of labeled connected graphs covering n vertices with at least one endpoint (vertex of degree 1).

Original entry on oeis.org

0, 0, 1, 3, 28, 475, 14646, 813813, 82060392, 15251272983, 5312295240010, 3519126783483377, 4487168285715524124, 11116496280631563128723, 53887232400918561791887118, 513757147287101157620965656285, 9668878162669182924093580075565776

Offset: 0

Gus Wiseman, Sep 04 2019

nonn

A graph is covering if the vertex set is the union of the edge set, so there are no isolated vertices.

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(4) = 28 connected covering graphs with at least one endpoint.

The non-connected version is A327227.
The non-covering version is A327364.
Graphs with endpoints are A245797.
Connected covering graphs are A001187.
Connected graphs with bridges are A327071.
Cf. A004110, A059166, A006125, A006129, A059166, A100743, A141580, A322395, A327105, A327229, A327230, A327366, A327369, A327377.

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]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Length[csm[#]]==1&&Min@@Length/@Split[Sort[Join@@#]]==1&]],{n,0,5}]

seq(n)={Vec(serlaplace(-x^2/2 + log(sum(k=0, n, 2^binomial(k, 2)*x^k/k! + O(x*x^n))) - log(sum(k=0, n, 2^binomial(k, 2)*(x*exp(-x + O(x^n)))^k/k!))), -(n+1))} \\ Andrew Howroyd, Sep 11 2019

Inverse binomial transform of A327364.

a(n) = A001187(n) - A059166(n). - Andrew Howroyd, Sep 11 2019

Terms a(7) and beyond from Andrew Howroyd, Sep 11 2019

A370064 Triangle read by rows: T(n,k) is the number of simple connected graphs on n labeled nodes with k articulation vertices, (0 <= k <= n).

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 3, 0, 0, 10, 16, 12, 0, 0, 238, 250, 180, 60, 0, 0, 11368, 8496, 4560, 1920, 360, 0, 0, 1014888, 540568, 211680, 75600, 21000, 2520, 0, 0, 166537616, 61672192, 17186624, 4663680, 1226400, 241920, 20160, 0, 0, 50680432112, 12608406288, 2416430016, 469336896, 98431200, 20109600, 2963520, 181440, 0, 0

Offset: 0

Andrew Howroyd, Feb 23 2024

			Triangle begins:
        1;
        1,      0;
        1,      0,      0;
        1,      3,      0,     0;
       10,     16,     12,     0,     0;
      238,    250,    180,    60,     0,    0;
    11368,   8496,   4560,  1920,   360,    0, 0;
  1014888, 540568, 211680, 75600, 21000, 2520, 0, 0;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
S. Selkow, The enumeration of labeled graphs by number of cutpoints, Discr. Math. 185 (1998), 183-191.

Columns k=0..3 are A013922(n>1), A013923, A013924, A013925.
Row sums are A001187.
Cf. A001710, A325111 (unlabeled version).

J(p, n)={my(u=Vecrev(p,1+n)); forstep(k=n, 1, -1, u[k] -= k*u[k+1]; u[k]/=n+1-k); u}
G(n)={log(x/serreverse(x*deriv(log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x*x^n)))))}
T(n)={my(v=Vec(serlaplace( 1 + ((y-1)*x + serreverse(x/((1-y) + y*exp(G(n)))))/y ))); vector(#v, n, J(v[n], n-1))}
{ my(A=T(7)); for(i=1, #A, print(A[i])) }

T(n, n-2) = n!/2 = A001710(n) for n >= 2.

A371445 Numbers whose distinct prime indices are binary carry-connected and have no binary containments.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 55, 59, 61, 64, 65, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 115, 121, 125, 127, 128, 131, 137, 139, 143, 145, 149, 151, 157, 163, 167, 169, 173, 179, 181

Offset: 1

Gus Wiseman, Mar 30 2024

nonn

Also Heinz numbers of binary carry-connected integer partitions whose distinct parts have no binary containments, counted by A371446.
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.
A binary carry of two positive integers is an overlap of binary indices. A multiset is said to be binary carry-connected iff the graph whose vertices are the elements and whose edges are binary carries is connected.
A binary containment is a containment of binary indices. For example, the numbers {3,5} have binary indices {{1,2},{1,3}}, so there is a binary carry but not a binary containment.

			The terms together with their prime indices begin:
     2: {1}            37: {12}              97: {25}
     3: {2}            41: {13}             101: {26}
     4: {1,1}          43: {14}             103: {27}
     5: {3}            47: {15}             107: {28}
     7: {4}            49: {4,4}            109: {29}
     8: {1,1,1}        53: {16}             113: {30}
     9: {2,2}          55: {3,5}            115: {3,9}
    11: {5}            59: {17}             121: {5,5}
    13: {6}            61: {18}             125: {3,3,3}
    16: {1,1,1,1}      64: {1,1,1,1,1,1}    127: {31}
    17: {7}            65: {3,6}            128: {1,1,1,1,1,1,1}
    19: {8}            67: {19}             131: {32}
    23: {9}            71: {20}             137: {33}
    25: {3,3}          73: {21}             139: {34}
    27: {2,2,2}        79: {22}             143: {5,6}
    29: {10}           81: {2,2,2,2}        145: {3,10}
    31: {11}           83: {23}             149: {35}
    32: {1,1,1,1,1}    89: {24}             151: {36}

Contains all powers of primes A000961 except 1.
Case of A325118 (counted by A325098) without binary containments.
For binary indices of binary indices we have A326750 = A326704 /\ A326749.
For prime indices of prime indices we have A329559 = A305078 /\ A316476.
An opposite version is A371294 = A087086 /\ A371291.
Partitions of this type are counted by A371446.
Carry-connected case of A371455 (counted by A325109).
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.
A326964 counts connected set-systems, covering A323818.
Cf. A019565, A056239, A112798, A304713, A304716, A305079, A305148, A325097, A325105, A325107, A325119, A371452.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
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]]]]]]]]];
Select[Range[100],stableQ[bpe/@prix[#],SubsetQ] && Length[csm[bpe/@prix[#]]]==1&]

Intersection of A371455 and A325118.

A371451 Number of connected components of the binary indices of the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Apr 01 2024

nonn

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.

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 binary indices of prime indices of 805 are {{1,2},{3},{1,4}}, with 2 connected components {{1,2},{1,4}} and {{3}}, so a(805) = 2.

For prime indices of prime indices we have A305079, ones A305078.
Positions of ones are A325118.
Positions of first appearances are A325782.
For prime indices of binary indices we have A371452, ones A371291.
For binary indices of binary indices we have A326753, ones A326749.
A001187 counts connected graphs.
A007718 counts non-isomorphic connected multiset partitions.
A048143 counts connected antichains of sets.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A070939 gives length of binary expansion.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A326964 counts connected set-systems, covering A323818.
Cf. A000720, A019565, A087086, A096111, A325097, A326782, A368109, A371292, A371294, A371445, A371447.

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]]]]]]]]];
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[csm[bix/@prix[n]]],{n,100}]

zero_first_elem_and_bitmask_connected_elems(ys) = { my(cs = List([ys[1]]), i=1); ys[1] = 0; while(i<=#cs, for(j=2, #ys, if(ys[j]&&(0!=bitand(cs[i], ys[j])), listput(cs, ys[j]); ys[j] = 0)); i++); (ys); };
A371451(n) = if(1==n, 0, my(cs = apply(p -> primepi(p), factor(n)[, 1]~), s=0); while(#cs, cs = select(c -> c, zero_first_elem_and_bitmask_connected_elems(cs)); s++); (s)); \\ Antti Karttunen, Jan 29 2025

Data section extended to a(105) by Antti Karttunen, Jan 29 2025

cp's OEIS Frontend

A327200 Number of labeled graphs with n vertices and non-spanning edge-connectivity >= 2.

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A370317 Number of labeled graphs with n vertices (allowing isolated vertices) and n edges, such that the edge set is connected.

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A002032 Number of n-colored connected graphs on n labeled nodes.

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A123527 Triangular array T(n,k) giving number of connected graphs with n labeled nodes and k edges (n >= 1, n-1 <= k <= n(n-1)/2).

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A129584 Number of unlabeled bi-point-determining graphs: graphs in which no two vertices have the same neighborhoods or the same augmented neighborhoods (the augmented neighborhood of a vertex is the neighborhood of the vertex union the vertex itself).

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A324464 Number of connected graphical necklaces with n vertices.

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A327362 Number of labeled connected graphs covering n vertices with at least one endpoint (vertex of degree 1).

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A370064 Triangle read by rows: T(n,k) is the number of simple connected graphs on n labeled nodes with k articulation vertices, (0 <= k <= n).

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