A304716 -id:A304716 - cp's OEIS frontend

A328513 Connected squarefree numbers.

1, 2, 3, 5, 7, 11, 13, 17, 19, 21, 23, 29, 31, 37, 39, 41, 43, 47, 53, 57, 59, 61, 65, 67, 71, 73, 79, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 127, 129, 131, 133, 137, 139, 149, 151, 157, 159, 163, 167, 173, 179, 181, 183, 185, 191, 193, 195

Offset: 1

Gus Wiseman, Oct 20 2019

nonn

First differs from A318718 and A318719 in having 195 = prime(2) * prime(3) * prime(6).

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 sequence of all connected sets of multisets together with their MM-numbers (A302242) begins:
   1: {}
   2: {{}}
   3: {{1}}
   5: {{2}}
   7: {{1,1}}
  11: {{3}}
  13: {{1,2}}
  17: {{4}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  23: {{2,2}}
  29: {{1,3}}
  31: {{5}}
  37: {{1,1,2}}
  39: {{1},{1,2}}
  41: {{6}}
  43: {{1,4}}
  47: {{2,3}}
  53: {{1,1,1,1}}
  57: {{1},{1,1,1}}

Gus Wiseman, Sequences counting and encoding certain classes of multisets

A subset of A005117.
These are Heinz numbers of the partitions counted by A304714.
The maximum connected squarefree divisor of n is A327398(n).
Cf. A056239, A112798, A286518, A302242, A304716, A305078, A305079, A327076, 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]]]]]]]]];
Select[Range[100],SquareFreeQ[#]&&Length[zsm[primeMS[#]]]<=1&]

Intersection of A005117 and A305078.

A371446 Number of carry-connected integer partitions whose distinct parts have no binary containments.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 2, 5, 4, 4, 4, 8, 4, 7, 7, 12, 10, 14, 12, 15, 19, 19, 21, 32, 27, 33, 40, 46, 47, 61, 52, 75, 89, 95, 104, 129, 129, 149, 176, 188, 208, 249, 257, 296, 341, 373, 394, 476, 496, 552

Offset: 0

Gus Wiseman, Apr 02 2024

These partitions are ranked by A371445.
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. An integer partition is binary carry-connected iff the graph with one vertex for each part and edges corresponding to 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 a(12) = 8 through a(14) = 7 partitions:
  (12)             (13)                         (14)
  (6,6)            (10,3)                       (7,7)
  (9,3)            (5,5,3)                      (9,5)
  (4,4,4)          (1,1,1,1,1,1,1,1,1,1,1,1,1)  (6,5,3)
  (6,3,3)                                       (5,3,3,3)
  (3,3,3,3)                                     (2,2,2,2,2,2,2)
  (2,2,2,2,2,2)                                 (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)

The first condition (carry-connected) is A325098.
The second condition (stable) is A325109.
Ranks for binary indices of binary indices are A326750 = A326704 /\ A326749.
Ranks for prime indices of prime indices are A329559 = A305078 /\ A316476.
Ranks for prime indices of binary indices are A371294 = A087086 /\ A371291.
Ranks for binary indices of prime indices are A371445 = A325118 /\ A371455.
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.
A326964 counts connected set-systems, covering A323818.
Cf. A019565, 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}];
bix[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]]]]]]]]];
Table[Length[Select[IntegerPartitions[n], stableQ[bix/@Union[#],SubsetQ]&&Length[csm[bix/@#]]<=1&]],{n,0,30}]

A305081 Heinz numbers of z-trees. Heinz numbers of connected integer partitions with pairwise indivisible parts and z-density -1.

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 91, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 203, 211, 223, 227, 229

Offset: 1

Gus Wiseman, May 25 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
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 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 clutter density of a multiset S of positive integers is Sum_{s in S} (omega(s) - 1) - omega(lcm(S)) where omega = A001221.

			4331 is the Heinz number of {18,20}, which is a z-tree corresponding to the multiset multisystem {{1,2,2},{1,1,3}}.
17927 is the Heinz number of {4,6,45}, which is a z-tree corresponding to the multiset multisystem {{1,1},{1,2},{2,2,3}}.
27391 is the Heinz number of {4,4,6,14}, which is a z-tree corresponding to the multiset multisystem {{1,1},{1,1},{1,2},{1,4}}.

Cf. A001221, A030019, A056239, A112798, A286520, A302242, A303362, A303837, A304118, A304714, A304716, A305052, A305078, A305079.

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]]]]]]]]];
zens[n_]:=If[n==1,0,Total@Cases[FactorInteger[n],{p_,k_}:>k*(PrimeNu[PrimePi[p]]-1)]-PrimeNu[LCM@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]]]];
Select[Range[300],And[zens[#]==-1,Length[zsm[primeMS[#]]]==1,Select[Tuples[primeMS[#],2],UnsameQ@@#&&Divisible@@#&]=={}]&]

A305253 Number of connected factorizations of n into factors greater than 1 whose distinct factors are pairwise indivisible.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 28 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 whose distinct factors are pairwise indivisible and such that G(S) is a connected graph.

			The a(360) = 8 factorizations: (360), (4*90), (10*36), (12*30), (15*24), (18*20), (4*6*15), (6*6*10).

Antti Karttunen, Table of n, a(n) for n = 1..20160
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A001970, A048143, A281116, A285572, A286518, A286520, A303386, A304714, A304716, A305149, A305150, A305193.

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]]}]];
sacs[n_]:=Select[facs[n],Function[f,Length[zsm[f]]==1&&Select[Tuples[Union[f],2],UnsameQ@@#&&Divisible@@#&]=={}]]
Table[Length[sacs[n]],{n,500}]

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)); };
A305253aux(n, m, facs) = if(1==n, is_connected(Vec(facs)), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m)&&factorback(apply(x -> (x==d)||(x%d),Vec(facs))), newfacs = List(facs); listput(newfacs,d); s += A305253aux(n/d, d, newfacs))); (s));
A305253(n) = if(1==n,0,A305253aux(n, n, List([]))); \\ Antti Karttunen, Dec 06 2018

a(n) <= A305193(n) <= A001055(n). - Antti Karttunen, Dec 06 2018

Definition clarified by Gus Wiseman, more terms from Antti Karttunen, Dec 06 2018

A329625 Smallest BII-number of a connected set-system with n edges.

Original entry on oeis.org

0, 1, 5, 7, 23, 31, 63, 127, 383, 511, 1023, 2047, 4095, 8191

Offset: 0

Gus Wiseman, Nov 28 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 of positive integers) 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 sequence of terms together with their corresponding set-systems begins:
     0: {}
     1: {{1}}
     5: {{1},{1,2}}
     7: {{1},{2},{1,2}}
    23: {{1},{2},{1,2},{1,3}}
    31: {{1},{2},{1,2},{3},{1,3}}
    63: {{1},{2},{1,2},{3},{1,3},{2,3}}
   127: {{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3}}
   383: {{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3},{1,4}}
   511: {{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3},{4},{1,4}}

The smallest BII-number of a set-system with n edges is A000225(n).
The smallest BII-number of a set-system with n vertices is A072639(n).
BII-numbers of connected set-systems are A326749.
MM-numbers of connected set-systems are A328514.
The case of clutters is A329627.
Cf. A048143, A048793, A070939, A304716, A305078, A326031, A326753, A329552, A329553, A329626.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
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]]]]]]]]];
First/@GatherBy[Select[Range[0,1000],Length[csm[bpe/@bpe[#]]]<=1&],Length[bpe[#]]&]

A305054 If n = Product_i prime(x_i)^k_i, then a(n) = Sum_i k_i * omega(x_i), where omega = A001221 is number of distinct prime factors.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 24 2018

nonn

Cf. A001221, A003963, A056239, A076078, A112798, A286520, A290103, A302242, A304714, A304716, A305052, A305053.

Table[If[n==1,0,Total@Cases[FactorInteger[n],{p_,k_}:>k*PrimeNu[PrimePi[p]]]],{n,100}]

a(n) = {my(f=factor(n)); sum(k=1, #f~, f[k,2]*omega(primepi(f[k,1])));} \\ Michel Marcus, Jun 09 2018

Totally additive with a(prime(n)) = omega(n).

a(n) = A305053(n) + A001221(n). - Michel Marcus, Jun 09 2018

A305501 Number of connected components of the integer partition y + 1 where y is the integer partition with Heinz number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 03 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
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 greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A partition y is said to be connected if G(U(y + 1)) is a connected graph, where U(y + 1) is the set of distinct successors of the parts of y.
This is intended to be a cleaner form of A305079, where the treatment of empty multisets is arbitrary.

			The "prime index plus 1" multiset of 7410 is {2,3,4,7,9}, with connected components {{2,4},{3,9},{7}}, so a(7410) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to Heinz numbers

Cf. A001221, A048143, A056239, A112798, A275024, A286518, A302242, A303837, A304118, A304714, A304716, A305078, A305079, A305504.

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[zsm[primeMS[n]+1]],{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); };
A305501(n) = { my(cs = apply(p -> 1+primepi(p),factor(n)[,1]~), s=0); while(#cs, cs = select(c -> c, zero_first_elem_and_connected_elems(cs)); s++); (s); }; \\ Antti Karttunen, Nov 09 2018

More terms from Antti Karttunen, Nov 09 2018

A305831 Number of connected components of the strict integer partition with FDH number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 10 2018

nonn

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 set S is said to be connected if G(S) is a connected graph.

			Let f = A050376. The FD-factorization of 1683 is 9*11*17 = f(6)*f(7)*f(10). The connected components of {6,7,10} are {{7},{6,10}}, so a(1683) = 2.

Cf. A048143, A050376, A064547, A213925, A299755, A299756, A304714, A304716, A305078, A305079, A305829, A305830, A305832.

FDfactor[n_]:=If[n===1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}->2^(m-1)]]]]];
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]]]]]]]]];
nn=200;FDprimeList=Array[FDfactor,nn,1,Union];FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
Table[Length[zsm[FDfactor[n]/.FDrules]],{n,nn}]

A305832 Number of connected components of the n-th FDH set-system.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 10 2018

nonn

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. Every positive integer n has a unique factorization of the form n = f(s_1)*...*f(s_k) where the s_i are strictly increasing positive integers. The n-th FDH set-system is obtained by repeating this factorization on each index s_i.

			Let f = A050376. The FD-factorization of 765 is 5*9*17 or f(4)*f(6)*f(10) = f(4)*f(2*3)*f(2*5) with connected components {{{4}},{{2,3},{2,5}}}, so a(765) = 2.

Cf. A048143, A050376, A064547, A213925, A299755, A299756, A304714, A304716, A305078, A305079, A305829, A305830, A305831.

FDfactor[n_]:=If[n===1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}->2^(m-1)]]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>1]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
nn=100;FDprimeList=Array[FDfactor,nn,1,Union];FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
Table[Length[csm[FDfactor[#]/.FDrules&/@(FDfactor[n]/.FDrules)]],{n,nn}]

A322367 Number of disconnected or empty integer partitions of n.

Original entry on oeis.org

1, 0, 1, 2, 3, 6, 7, 14, 17, 27, 34, 54, 63, 98, 118, 165, 207, 287, 345, 474, 574, 757, 931, 1212, 1463, 1890, 2292, 2898, 3515, 4413, 5303

Offset: 0

Gus Wiseman, Dec 04 2018

An integer partition is connected if the prime factorizations of its parts form a connected hypergraph. It is disconnected if it can be separated into two or more integer partitions with relatively prime products. For example, the integer partition (654321) has three connected components: (6432)(5)(1).

			The a(3) = 2 through a(9) = 27 disconnected integer partitions:
  (21)   (31)    (32)     (51)      (43)       (53)        (54)
  (111)  (211)   (41)     (321)     (52)       (71)        (72)
         (1111)  (221)    (411)     (61)       (332)       (81)
                 (311)    (2211)    (322)      (431)       (432)
                 (2111)   (3111)    (331)      (521)       (441)
                 (11111)  (21111)   (421)      (611)       (522)
                          (111111)  (511)      (3221)      (531)
                                    (2221)     (3311)      (621)
                                    (3211)     (4211)      (711)
                                    (4111)     (5111)      (3222)
                                    (22111)    (22211)     (3321)
                                    (31111)    (32111)     (4221)
                                    (211111)   (41111)     (4311)
                                    (1111111)  (221111)    (5211)
                                               (311111)    (6111)
                                               (2111111)   (22221)
                                               (11111111)  (32211)
                                                           (33111)
                                                           (42111)
                                                           (51111)
                                                           (222111)
                                                           (321111)
                                                           (411111)
                                                           (2211111)
                                                           (3111111)
                                                           (21111111)
                                                           (111111111)

Cf. A054921, A218970, A286518, A322335, A304714, A304716, A305078, A305079, A322306, A322307, A322337, A322338, A322368, A322369.

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[IntegerPartitions[n],Length[zsm[#]]!=1&]],{n,20}]

cp's OEIS Frontend

A328513 Connected squarefree numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A371446 Number of carry-connected integer partitions whose distinct parts have no binary containments.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A305081 Heinz numbers of z-trees. Heinz numbers of connected integer partitions with pairwise indivisible parts and z-density -1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A305253 Number of connected factorizations of n into factors greater than 1 whose distinct factors are pairwise indivisible.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A329625 Smallest BII-number of a connected set-system with n edges.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A305054 If n = Product_i prime(x_i)^k_i, then a(n) = Sum_i k_i * omega(x_i), where omega = A001221 is number of distinct prime factors.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A305501 Number of connected components of the integer partition y + 1 where y is the integer partition with Heinz number n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A305831 Number of connected components of the strict integer partition with FDH number n.

Views

Author

Keywords

Comments

Examples

Crossrefs