A285572 -id:A285572 - cp's OEIS frontend

A343661 Sum of numbers of y-multisets of divisors of x for each x >= 1, y >= 0, x + y = n.

1, 2, 4, 7, 12, 19, 30, 46, 70, 105, 155, 223, 316, 443, 619, 865, 1210, 1690, 2354, 3263, 4497, 6157, 8368, 11280, 15078, 19989, 26296, 34356, 44626, 57693, 74321, 95503, 122535, 157101, 201377, 258155, 330994, 424398, 544035, 696995, 892104, 1140298, 1455080

Offset: 1

Gus Wiseman, Apr 30 2021

nonn

			The a(5) = 12 multisets of divisors:
  {1,1,1,1}  {1,1,1}  {1,1}  {1}  {}
             {1,1,2}  {1,3}  {2}
             {1,2,2}  {3,3}  {4}
             {2,2,2}

Antidiagonal sums of the array A343658 (or row sums of the triangle).
Dominates A343657.
A000005 counts divisors.
A007318 counts k-sets of elements of {1..n}.
A059481 counts k-multisets of elements of {1..n}.
A343656 counts divisors of powers.
Cf. A000169, A000312, A009998, A062319, A067824, A143773, A146291, A176029, A184389, A285572, A326077, A327527, A334996, A343652, A343657.

multchoo[n_,k_]:=Binomial[n+k-1,k];
Table[Sum[multchoo[DivisorSigma[0,k],n-k],{k,n}],{n,10}]

a(n) = Sum_{k=1..n} binomial(sigma(k) + n - k - 1, n - k).

A096826 Number of maximal-sized antichains in divisor lattice D(n).

Original entry on oeis.org

1, 2, 2, 3, 2, 1, 2, 4, 3, 1, 2, 3, 2, 1, 1, 5, 2, 3, 2, 3, 1, 1, 2, 6, 3, 1, 4, 3, 2, 2, 2, 6, 1, 1, 1, 1, 2, 1, 1, 6, 2, 2, 2, 3, 3, 1, 2, 10, 3, 3, 1, 3, 2, 6, 1, 6, 1, 1, 2, 1, 2, 1, 3, 7, 1, 2, 2, 3, 1, 2, 2, 4, 2, 1, 3, 3, 1, 2, 2, 10, 5, 1, 2, 1, 1, 1

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 17 2004

nonn

The divisor lattice D(n) is the lattice of the divisors of the natural number n.

			From _Gus Wiseman_, Aug 24 2018: (Start)
The a(120) = 6 antichains:
  {8,12,20,30}
  {8,12,15,20}
  {8,10,12,15}
  {6,8,15,20}
  {6,8,10,15}
  {4,6,10,15}
(End)

Eric M. Schmidt, Table of n, a(n) for n = 1..10000

Cf. A000005, A008480, A096825, A096827, A253249, A285572 A285573.

def A096826(n) :
    if n==1 : return 1
    R. = QQ[]; mults = [x[1] for x in factor(n)]
    maxsize = prod((t^(m+1)-1)//(t-1) for m in mults)[sum(mults)//2]
    dlat = LatticePoset((divisors(n), attrcall("divides")))
    count = 0
    for ac in dlat.antichains_iterator() :
        if len(ac) == maxsize : count += 1
    return count
# Eric M. Schmidt, May 13 2013

More terms from Eric M. Schmidt, May 13 2013

A305567 Irregular triangle where T(n,k) is the number of finite sets of positive integers with least common multiple n and greatest common divisor k, where k runs over all divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 4, 2, 1, 1, 2, 1, 1, 7, 1, 1, 1, 1, 1, 32, 7, 2, 1, 1, 1, 1, 1, 7, 1, 1, 1, 7, 1, 1, 1, 8, 4, 2, 1, 1, 1, 1, 32, 2, 7, 1, 1, 1, 1, 1, 32, 7, 1, 2, 1, 1, 7, 1, 1, 1, 7, 1, 1, 1, 1, 1, 136, 32, 4, 7, 2, 1, 1, 1, 2

Offset: 1

Gus Wiseman, Jun 05 2018

			Triangle begins:
   1
   1  1
   1  1
   2  1  1
   1  1
   7  1  1  1
   1  1
   4  2  1  1
   2  1  1
   7  1  1  1
   1  1
  32  7  2  1  1  1
   1  1
   7  1  1  1
   7  1  1  1
   8  4  2  1  1
   1  1
  32  2  7  1  1  1
   1  1
  32  7  1  2  1  1

Row lengths are A000005.

Cf. A076078, A181819, A281116, A285572, A290103, A304818, A305563, A305564, A305565, A305566.

Table[Length[Select[Subsets[Divisors[n]],And[GCD@@#==k,LCM@@#==n]&]],{n,30},{k,Divisors[n]}]

A305106 Number of unitary factorizations of Heinz numbers of integer partitions of n. Number of multiset partitions of integer partitions of n with pairwise disjoint blocks.

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 21, 34, 55, 87, 138, 211, 324, 486, 727, 1079, 1584, 2305, 3337, 4789, 6830, 9712, 13689, 19225, 26841, 37322, 51598, 71108, 97580, 133350, 181558, 246335, 332991, 448706, 602607, 806732, 1077333, 1433885, 1903682, 2520246, 3328549, 4383929

Offset: 0

Gus Wiseman, May 25 2018

nonn

			The a(6) = 21 unitary factorizations:
(13) (21) (22) (25) (27) (28) (30) (36) (40) (48) (64)
(2*11) (2*15) (3*7) (3*10) (3*16) (4*7) (4*9) (5*6) (5*8)
(2*3*5)
The a(6) = 21 multiset partitions:
{{6}}
{{2,4}}
{{1,5}}
{{3,3}}
{{2,2,2}}
{{1,1,4}}
{{1,2,3}}
{{1,1,2,2}}
{{1,1,1,3}}
{{1,1,1,1,2}}
{{1,1,1,1,1,1}}
{{1},{5}}
{{1},{2,3}}
{{2},{4}}
{{2},{1,3}}
{{2},{1,1,1,1}}
{{1,1},{4}}
{{1,1},{2,2}}
{{3},{1,2}}
{{3},{1,1,1}}
{{1},{2},{3}}

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000110, A001055, A001221, A001970, A034444, A089233, A258466, A259936, A281116, A285572, A305078, A305079.

Row sums of A321878.

Table[Sum[BellB[Length[Union[y]]],{y,IntegerPartitions[n]}],{n,30}]
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[With[{t = n - i j}, b[t, Min[t, i - 1], k]], {j, 1, n/i}] k + b[n, i - 1, k]]];
T[n_, k_] := Sum[b[n, n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]/k!;
a[n_] := Sum[T[n, k], {k, 0, Floor[(Sqrt[1 + 8n] - 1)/2]}];
a /@ Range[0, 50] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz in A321878 *)

A305194 Number of z-forests summing to n. Number of strict integer partitions of n with pairwise indivisible parts and all connected components having clutter density -1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 3, 4, 5, 4, 6, 7, 7, 9, 11, 12, 13, 15, 17, 20, 23, 25, 27, 32, 35, 40, 45, 49, 54, 58, 67, 78, 82, 95, 99, 111, 123, 135, 150, 164, 177, 194, 214, 236, 260, 282, 309, 330

Offset: 1

Gus Wiseman, May 27 2018

Given a finite set S of positive integers greater than 1, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices that have a common divisor greater than 1. 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. The clutter density of S is defined to be Sum_{s in S} (omega(s) - 1) - omega(lcm(S)), where omega = A001221 and lcm is least common multiple. Then a z-forest is a strict integer partition with pairwise indivisible parts greater than 1 such that all connected components have clutter density -1.

			The a(17) = 11 z-forests together with the corresponding multiset systems:
       (17): {{7}}
     (15,2): {{2,3},{1}}
     (14,3): {{1,4},{2}}
     (13,4): {{6},{1,1}}
     (12,5): {{1,1,2},{3}}
     (11,6): {{5},{1,2}}
     (10,7): {{1,3},{4}}
      (9,8): {{2,2},{1,1,1}}
   (10,4,3): {{1,3},{1,1},{2}}
    (7,6,4): {{4},{1,2},{1,1}}
  (7,5,3,2): {{4},{3},{2},{1}}

Roland Bacher, On the enumeration of labelled hypertrees and of labelled bipartite trees, arXiv:1102.2708 [math.CO], 2011.

Cf. A030019, A048143, A134954, A275307, A285572, A293510, A293993, A293994, A303362, A303837, A303838, A304118, A304382, A305078, A305195.

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]]]]]]]]];
zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
zreeQ[s_]:=And[Length[s]>=2,zensity[s]==-1];
Table[Length[Select[IntegerPartitions[n],Function[s,UnsameQ@@s&&And@@(Length[#]==1||zreeQ[#]&)/@Table[Select[s,Divisible[m,#]&],{m,zsm[s]}]&&Select[Tuples[s,2],UnsameQ@@#&&Divisible@@#&]=={}]]],{n,50}]

A305195 Number of z-blobs summing to n. Number of connected strict integer partitions of n, with pairwise indivisible parts, that cannot be capped by a z-tree.

Original entry on oeis.org

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, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 2, 1, 1, 1, 3, 2, 2, 2, 1, 1, 3, 3, 3, 1, 1, 1, 4, 5, 6, 2, 1, 1, 4, 6, 7, 2, 2, 6

Offset: 1

Gus Wiseman, May 27 2018

nonn

Caps of a clutter are defined in the link, and the generalization to "multiclutters," where edges can be multisets, is straightforward.

			The a(30) = 2 z-blobs together with the corresponding multiset systems:
     (30): {{1,2,3}}
  (18,12): {{1,2,2},{1,1,2}}
The a(47) = 3 z-blobs together with the corresponding multiset systems:
        (47): {{15}}
  (21,14,12): {{2,4},{1,4},{1,1,2}}
  (20,15,12): {{1,1,3},{2,3},{1,1,2}}
The a(60) = 5 z-blobs together with the corresponding multiset systems:
           (60): {{1,1,2,3}}
        (42,18): {{1,2,4},{1,2,2}}
        (36,24): {{1,1,2,2},{1,1,1,2}}
     (30,18,12): {{1,2,3},{1,2,2},{1,1,2}}
  (21,15,14,10): {{2,4},{2,3},{1,4},{1,3}}
The a(67) = 7 z-blobs together with the corresponding multiset systems:
           (67): {{19}}
     (45,12,10): {{2,2,3},{1,1,2},{1,3}}
     (42,15,10): {{1,2,4},{2,3},{1,3}}
     (40,15,12): {{1,1,1,3},{2,3},{1,1,2}}
     (33,22,12): {{2,5},{1,5},{1,1,2}}
     (28,21,18): {{1,1,4},{2,4},{1,2,2}}
  (24,18,15,10): {{1,1,1,2},{1,2,2},{2,3},{1,3}}

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

Cf. A030019, A048143, A275307, A285572, A293510, A303362, A303837, A303838, A304118, A304382, A304887, A305028, A305078, A305194.

A317101 Numbers whose prime multiplicities are pairwise indivisible.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 77, 78, 79, 81, 82, 83, 85, 86, 87

Offset: 1

Gus Wiseman, Aug 01 2018

nonn

			72 = 2^3 * 3^2 is in the sequence because 3 and 2 are pairwise indivisible.

Cf. A118914, A124010, A285572, A285573, A303362, A304713, A316475, A317102, A317616.

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

A318401 Numbers whose prime indices are distinct and pairwise indivisible and whose own prime indices span an initial interval of positive integers.

Original entry on oeis.org

1, 2, 3, 7, 13, 15, 19, 35, 37, 53, 61, 69, 89, 91, 95, 113, 131, 141, 143, 145, 151, 161, 165, 223, 247, 251, 265, 281, 299, 309, 311, 329, 355, 359, 377, 385, 407, 427, 437, 463, 503, 591, 593, 611, 655, 659, 667, 671, 689, 703, 719, 721, 759, 791, 827, 851

Offset: 1

Gus Wiseman, Dec 16 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}}. This sequence lists all MM-numbers of strict antichains of multisets spanning an initial interval of positive integers.

			The sequence of multisystems whose MM-numbers belong to the sequence begins:
   1: {}
   2: {{}}
   3: {{1}}
   7: {{1,1}}
  13: {{1,2}}
  15: {{1},{2}}
  19: {{1,1,1}}
  35: {{2},{1,1}}
  37: {{1,1,2}}
  53: {{1,1,1,1}}
  61: {{1,2,2}}
  69: {{1},{2,2}}
  89: {{1,1,1,2}}
  91: {{1,1},{1,2}}
  95: {{2},{1,1,1}}

Cf. A003963, A006126, A055932, A056239, A112798, A285572, A290103, A293993, A302242, A304713, A316476, A319496, A319721, A319837, A320275, A320456.

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]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[200],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],stableQ[primeMS[#],Divisible]]&]

A321680 Number of non-isomorphic weight-n connected antichains (not necessarily strict) of multisets with multiset density -1.

Original entry on oeis.org

1, 1, 3, 4, 9, 14, 39, 80, 216, 538, 1460

Offset: 0

Gus Wiseman, Nov 16 2018

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.

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(1) = 1 through a(5) = 14 multiset trees:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}        {{1,1,1,1,1}}
         {{1,2}}    {{1,2,2}}      {{1,1,2,2}}        {{1,1,2,2,2}}
         {{1},{1}}  {{1,2,3}}      {{1,2,2,2}}        {{1,2,2,2,2}}
                    {{1},{1},{1}}  {{1,2,3,3}}        {{1,2,2,3,3}}
                                   {{1,2,3,4}}        {{1,2,3,3,3}}
                                   {{1,1},{1,1}}      {{1,2,3,4,4}}
                                   {{1,2},{2,2}}      {{1,2,3,4,5}}
                                   {{1,3},{2,3}}      {{1,1},{1,2,2}}
                                   {{1},{1},{1},{1}}  {{1,2},{2,2,2}}
                                                      {{1,2},{2,3,3}}
                                                      {{1,3},{2,3,3}}
                                                      {{1,4},{2,3,4}}
                                                      {{3,3},{1,2,3}}
                                                      {{1},{1},{1},{1},{1}}

Cf. A006126, A007718, A056156, A096827, A285572, A293993, A293994, A305052, A319719, A319721, A321194, A321585, A321681.

A321681 Number of non-isomorphic weight-n connected strict antichains of multisets with multiset density -1.

Original entry on oeis.org

1, 1, 2, 3, 7, 13, 35, 77, 205, 517, 1399

Offset: 0

Gus Wiseman, Nov 16 2018

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.

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(1) = 1 through a(5) = 13 trees:
  {{1}}  {{1,1}}  {{1,1,1}}  {{1,1,1,1}}    {{1,1,1,1,1}}
         {{1,2}}  {{1,2,2}}  {{1,1,2,2}}    {{1,1,2,2,2}}
                  {{1,2,3}}  {{1,2,2,2}}    {{1,2,2,2,2}}
                             {{1,2,3,3}}    {{1,2,2,3,3}}
                             {{1,2,3,4}}    {{1,2,3,3,3}}
                             {{1,2},{2,2}}  {{1,2,3,4,4}}
                             {{1,3},{2,3}}  {{1,2,3,4,5}}
                                            {{1,1},{1,2,2}}
                                            {{1,2},{2,2,2}}
                                            {{1,2},{2,3,3}}
                                            {{1,3},{2,3,3}}
                                            {{1,4},{2,3,4}}
                                            {{3,3},{1,2,3}}

Cf. A006126, A007718, A056156, A096827, A285572, A293993, A293994, A305052, A319557, A319565, A319719, A319721, A321585, A321680.

cp's OEIS Frontend

A343661 Sum of numbers of y-multisets of divisors of x for each x >= 1, y >= 0, x + y = n.

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A096826 Number of maximal-sized antichains in divisor lattice D(n).

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A305567 Irregular triangle where T(n,k) is the number of finite sets of positive integers with least common multiple n and greatest common divisor k, where k runs over all divisors of n.

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Mathematica

A305106 Number of unitary factorizations of Heinz numbers of integer partitions of n. Number of multiset partitions of integer partitions of n with pairwise disjoint blocks.

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Mathematica

A305194 Number of z-forests summing to n. Number of strict integer partitions of n with pairwise indivisible parts and all connected components having clutter density -1.

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A305195 Number of z-blobs summing to n. Number of connected strict integer partitions of n, with pairwise indivisible parts, that cannot be capped by a z-tree.

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A317101 Numbers whose prime multiplicities are pairwise indivisible.

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Mathematica

A318401 Numbers whose prime indices are distinct and pairwise indivisible and whose own prime indices span an initial interval of positive integers.

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Mathematica

A321680 Number of non-isomorphic weight-n connected antichains (not necessarily strict) of multisets with multiset density -1.

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