A285572 -id:A285572 - cp's OEIS frontend

A317074 Number of antichains of multisets with multiset-join a strongly normal multiset of size n.

1, 1, 3, 13, 148, 7685

Offset: 0

Gus Wiseman, Jul 20 2018

An antichain of multisets is a finite set of finite nonempty multisets, none of which is a submultiset of any other. A multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities. The multiset-join of a multiset system has the same vertices with multiplicities equal to the maxima of the multiplicities in the edges.

			The a(3) = 13 antichains of multisets:
  (111),
  (112), (11)(12), (2)(11),
  (123), (13)(23), (12)(23), (12)(13), (12)(13)(23), (3)(12), (2)(13), (1)(23), (1)(2)(3).

Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, Journal of Integer Sequences, Vol. 7 (2004).

Cf. A048143, A285572, A293993, A293994, A303837, A303838, A304998, A305001.

Cf. A317073, A317076, A317078, A317079.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
multijoin[mss__]:=Join@@Table[Table[x,{Max[Count[#,x]&/@{mss}]}],{x,Union[mss]}];
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
auu[m_]:=Select[stableSets[Union[Rest[Subsets[m]]],submultisetQ],multijoin@@#==m&];
Table[Length[Join@@Table[auu[m],{m,strnorm[n]}]],{n,5}]

A317616 Numbers whose prime multiplicities are not pairwise indivisible.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192

Offset: 1

Gus Wiseman, Aug 01 2018

nonn

The numbers of terms that do not exceed 10^k, for k = 2, 3, ..., are 26, 344, 3762, 38711, 390527, 3915874, 39192197, 392025578, 3920580540, ... . Apparently, the asymptotic density of this sequence exists and equals 0.392... . - Amiram Eldar, Sep 25 2024

			72 = 2^3 * 3^2 is not in the sequence because 3 and 2 are pairwise indivisible.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A118914, A124010, A285572, A285573, A303362, A304713, A316475, A317101, A317102.

Select[Range[100],!Select[Tuples[Last/@FactorInteger[#],2],And[UnsameQ@@#,Divisible@@#]&]=={}&]

is(k) = if(k == 1, 0, my(e = Set(factor(k)[,2])); if(vecmax(e) == 1, 0, for(i = 1, #e, for(j = 1, i-1, if(!(e[i] % e[j]), return(1)))); 0)); \\ Amiram Eldar, Sep 25 2024

A303364 Number of strict integer partitions of n with pairwise indivisible and squarefree parts.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 2, 1, 1, 3, 2, 2, 4, 3, 3, 4, 6, 5, 5, 6, 7, 8, 9, 10, 10, 11, 11, 14, 14, 17, 16, 18, 19, 23, 24, 27, 29, 30, 33, 36, 41, 41, 42, 46, 51, 56, 60, 66, 67, 71, 81, 86, 93, 96, 101, 110, 121, 129, 135, 144, 153, 159, 173, 192, 204, 207, 224

Offset: 1

Gus Wiseman, Apr 22 2018

nonn

			The a(23) = 9 strict integer partitions are (23), (13,10), (17,6), (21,2), (10,7,6), (11,7,5), (13,7,3), (11,7,3,2), (13,5,3,2).

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..700 (terms 0..400 from Andrew Howroyd)

Cf. A000009, A000837, A003238, A005117, A006126, A051424, A073576, A285572, A285573, A293606, A293993, A303362, A303365.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@SquareFreeQ/@#&&Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]==={}&]],{n,60}]

lista(nn)={local(Cache=Map());
  my(excl=vector(nn, n, sumdiv(n, d, 2^(n-d))));
  my(c(n, m, b)=
     if(n==0, 1,
        while(m>n || bittest(b,0), m--; b>>=1);
        my(hk=[n, m, b], z);
        if(!mapisdefined(Cache, hk, &z),
          z = if(m, self()(n, m-1, b>>1) + self()(n-m, m, bitor(b, excl[m])), 0);
          mapput(Cache, hk, z)); z));
  my(a(n)=c(n, n, sum(i=1, n, if(!issquarefree(i), 2^(n-i)))));
  for(n=1, nn, print1(a(n), ", "))
} \\ Andrew Howroyd, Nov 02 2019

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

A305564 Number of finite sets of relatively prime positive integers with least common multiple n.

Original entry on oeis.org

1, 1, 1, 2, 1, 7, 1, 4, 2, 7, 1, 32, 1, 7, 7, 8, 1, 32, 1, 32, 7, 7, 1, 136, 2, 7, 4, 32, 1, 193, 1, 16, 7, 7, 7, 322, 1, 7, 7, 136, 1, 193, 1, 32, 32, 7, 1, 560, 2, 32, 7, 32, 1, 136, 7, 136, 7, 7, 1, 3464, 1, 7, 32, 32, 7, 193, 1, 32, 7, 193, 1, 2852, 1, 7

Offset: 1

Gus Wiseman, Jun 05 2018

nonn

			The a(6) = 7 sets are {1,6}, {2,3}, {1,2,3}, {1,2,6}, {1,3,6}, {2,3,6}, {1,2,3,6}.

Cf. A001055, A071625, A076078, A181819, A275870, A281116, A285573, A285572, A290103, A304818, A305563, A305565, A305566, A305567.

Table[Length[Select[Rest[Subsets[Divisors[n]]],And[GCD@@#==1,LCM@@#==n]&]],{n,100}]

A305565 Regular triangle where T(n,k) is the number of finite sets of positive integers with least common multiple n and greatest common divisor k.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 0, 0, 0, 1, 7, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 4, 2, 0, 1, 0, 0, 0, 1, 2, 0, 1, 0, 0, 0, 0, 0, 1, 7, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 32, 7, 2, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Jun 05 2018

			The T(12,2) = 7 sets are {2,12}, {4,6}, {2,4,6}, {2,4,12}, {2,6,12}, {4,6,12}, {2,4,6,12}.
Triangle begins:
   1
   1  1
   1  0  1
   2  1  0  1
   1  0  0  0  1
   7  1  1  0  0  1
   1  0  0  0  0  0  1
   4  2  0  1  0  0  0  1
   2  0  1  0  0  0  0  0  1
   7  1  0  0  1  0  0  0  0  1
   1  0  0  0  0  0  0  0  0  0  1
  32  7  2  1  0  1  0  0  0  0  0  1

Cf. A076078, A181819, A281116, A285572, A290103, A304818, A305563, A305564, A305566, A305567.

Table[Length[Select[Subsets[Divisors[n]],And[GCD@@#==k,LCM@@#==n]&]],{n,20},{k,n}]

If k divides n then T(n,k) = T(n/k,1) = A305564(n/k); otherwise T(n,k) = 0.

A316438 Heinz numbers of integer partitions whose product is strictly greater than the LCM of the parts.

Original entry on oeis.org

9, 18, 21, 25, 27, 36, 39, 42, 45, 49, 50, 54, 57, 63, 65, 72, 75, 78, 81, 84, 87, 90, 91, 98, 99, 100, 105, 108, 111, 114, 115, 117, 121, 125, 126, 129, 130, 133, 135, 144, 147, 150, 153, 156, 159, 162, 168, 169, 171, 174, 175, 180, 182, 183, 185, 189, 195

Offset: 1

Gus Wiseman, Jul 03 2018

nonn

Also numbers n > 1 such that A290104(n) > 1.

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

			Sequence of partitions whose product is greater than their LCM begins: (22), (221), (42), (33), (222), (2211), (62), (421), (322), (44), (331), (2221), (82), (422), (63), (22111), (332), (621), (2222), (4211).

Cf. A056239, A074761, A143773, A285572, A289508, A290103, A296150, A316429, A316431, A316433, A316437.

Select[Range[2,300],With[{pms=Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]},Times@@pms/LCM@@pms>1]&]

A317079 Number of unlabeled antichains of multisets with multiset-join a multiset of size n.

Original entry on oeis.org

1, 1, 3, 9, 46, 450

Offset: 0

Gus Wiseman, Jul 20 2018

An antichain of multisets is a finite set of finite nonempty multisets, none of which is a submultiset of any other. The multiset-join of a multiset system has the same vertices with multiplicities equal to the maxima of the multiplicities in the edges.

			Non-isomorphic representatives of the a(3) = 9 antichains of multisets:
  (111),
  (122), (1)(22), (12)(22),
  (123), (1)(23), (13)(23), (1)(2)(3), (12)(13)(23).

Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, Journal of Integer Sequences, Vol. 7 (2004).

Cf. A007716, A255906, A261006, A285572, A293993, A293994, A304998.

Cf. A317073, A317074, A317075, A317076, A317080.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
multijoin[mss__]:=Join@@Table[Table[x,{Max[Count[#,x]&/@{mss}]}],{x,Union[mss]}]
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
auu[m_]:=Select[stableSets[Union[Rest[Subsets[m]]],submultisetQ],multijoin@@#==m&];
sysnorm[m_]:=First[Sort[sysnorm[m,1]]];sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
Table[Length[Union[sysnorm/@Join@@Table[auu[m],{m,strnorm[n]}]]],{n,5}]

A320799 Number of non-isomorphic (not necessarily strict) antichains of multisets of weight n with no singletons or leaves (vertices that appear only once).

Original entry on oeis.org

1, 0, 1, 1, 5, 4, 22, 27, 107, 212, 689

Offset: 0

Gus Wiseman, Nov 02 2018

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(2) = 1 through a(7) = 27 multiset partitions:
  {{11}}  {{111}}  {{1111}}    {{11111}}    {{111111}}      {{1111111}}
                   {{1122}}    {{11222}}    {{111222}}      {{1112222}}
                   {{11}{11}}  {{11}{122}}  {{112222}}      {{1122222}}
                   {{11}{22}}  {{11}{222}}  {{112233}}      {{1122333}}
                   {{12}{12}}               {{111}{111}}    {{111}{1222}}
                                            {{11}{1222}}    {{11}{12222}}
                                            {{111}{222}}    {{111}{2222}}
                                            {{112}{122}}    {{11}{12233}}
                                            {{11}{2222}}    {{111}{2233}}
                                            {{112}{222}}    {{112}{1222}}
                                            {{11}{2233}}    {{11}{22222}}
                                            {{112}{233}}    {{112}{2222}}
                                            {{122}{122}}    {{11}{22333}}
                                            {{123}{123}}    {{112}{2333}}
                                            {{11}{11}{11}}  {{113}{2233}}
                                            {{11}{12}{22}}  {{122}{1233}}
                                            {{11}{22}{22}}  {{222}{1122}}
                                            {{11}{22}{33}}  {{11}{11}{122}}
                                            {{11}{23}{23}}  {{11}{11}{222}}
                                            {{12}{12}{12}}  {{11}{12}{222}}
                                            {{12}{12}{22}}  {{11}{12}{233}}
                                            {{12}{13}{23}}  {{11}{22}{233}}
                                                            {{11}{22}{333}}
                                                            {{12}{12}{222}}
                                                            {{12}{12}{233}}
                                                            {{12}{12}{333}}
                                                            {{12}{13}{233}}

Cf. A006126, A096827, A285572, A293993, A293994, A302545, A318099, A319560, A319719, A319721.

Cf. A320797, A320798, A320804, A320811, A320812, A320813.

A333492 Position of first appearance of n in A271410 (LCM of binary indices).

Original entry on oeis.org

1, 2, 4, 8, 16, 6, 64, 128, 256, 18, 1024, 12, 4096, 66, 20, 32768, 65536, 258, 262144, 24, 68, 1026, 4194304, 132, 16777216, 4098, 67108864, 72, 268435456, 22, 1073741824, 2147483648, 1028, 65538, 80, 264, 68719476736, 262146, 4100, 144, 1099511627776, 70, 4398046511104

Offset: 1

Gus Wiseman, Mar 28 2020

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.

			The sequence together with the corresponding binary expansions and binary indices begins:
      1:                 1 ~ {1}
      2:                10 ~ {2}
      4:               100 ~ {3}
      8:              1000 ~ {4}
     16:             10000 ~ {5}
      6:               110 ~ {2,3}
     64:           1000000 ~ {7}
    128:          10000000 ~ {8}
    256:         100000000 ~ {9}
     18:             10010 ~ {2,5}
   1024:       10000000000 ~ {11}
     12:              1100 ~ {3,4}
   4096:     1000000000000 ~ {13}
     66:           1000010 ~ {2,7}
     20:             10100 ~ {3,5}
  32768:  1000000000000000 ~ {16}
  65536: 10000000000000000 ~ {17}
    258:         100000010 ~ {2,9}

Giovanni Resta, Table of n, a(n) for n = 1..1000

The version for prime indices is A330225.
The version for standard compositions is A333225.
Let q(k) be the binary indices of k:
- The sum of q(k) is A029931(k).
- The elements of q(k) are row k of A048793.
- The product of q(k) is A096111(k).
- The LCM of q(k) is A271410(k).
- The GCD of q(k) is A326674(k).
GCD of prime indices is A289508.
LCM of prime indices is A290103.
LCM of standard compositions is A333226.
Cf. A000120, A066099, A070939, A074761, A076078, A124767, A285572, A324837, A328219, A328451, A331579, A333227.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
q=Table[LCM@@bpe[n],{n,10000}];
Table[Position[q,i][[1,1]],{i,First[Split[Union[q],#1+1==#2&]]}]

Terms a(23) and beyond from Giovanni Resta, Mar 29 2020

cp's OEIS Frontend

A317074 Number of antichains of multisets with multiset-join a strongly normal multiset of size n.

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A317616 Numbers whose prime multiplicities are not pairwise indivisible.

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A303364 Number of strict integer partitions of n with pairwise indivisible and squarefree parts.

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

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A305564 Number of finite sets of relatively prime positive integers with least common multiple n.

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A305565 Regular triangle where T(n,k) is the number of finite sets of positive integers with least common multiple n and greatest common divisor k.

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A316438 Heinz numbers of integer partitions whose product is strictly greater than the LCM of the parts.

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A317079 Number of unlabeled antichains of multisets with multiset-join a multiset of size n.

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