A285573 -id:A285573 - cp's OEIS frontend

A316475 Number of locally stable rooted trees with n nodes, meaning no branch is a submultiset of any other (unequal) branch of the same root.

1, 1, 2, 3, 5, 7, 14, 25, 50, 101, 207, 426, 902, 1917, 4108, 8887, 19335, 42330, 93130, 205894, 456960, 1018098, 2275613, 5102248, 11471107, 25856413

Offset: 1

Gus Wiseman, Jul 04 2018

			The a(6) = 7 locally stable rooted trees:
(((((o)))))
((((oo))))
(((ooo)))
(((o)(o)))
((oooo))
((o)((o)))
(ooooo)

Gus Wiseman, The a(8) = 25 locally stable rooted trees with 8 nodes.

Cf. A000081, A285572, A285573, A303362, A304713, A316468, A316470, A316473, A316474.

submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]]
strut[n_]:=strut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[strut/@c]]]/@IntegerPartitions[n-1],Select[Tuples[#,2],UnsameQ@@#&&submultisetQ@@#&]=={}&]];
Table[Length[strut[n]],{n,15}]

a(21)-a(26) from Robert Price, Sep 13 2018

A319719 Number of non-isomorphic connected antichains of multisets of weight n.

Original entry on oeis.org

1, 1, 3, 4, 10, 14, 48, 95, 305, 822, 2615

Offset: 0

Gus Wiseman, Sep 26 2018

In an antichain, no part is a proper submultiset of any other. The weight of an antichain is the sum of sizes of its parts. Weight is generally not the same as number of vertices. Connected antichains are also called clutters.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 10 connected antichains:
1: {{1}}
2: {{1,1}}
   {{1,2}}
   {{1},{1}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1,2,3}}
   {{1},{1},{1}}
4: {{1,1,1,1}}
   {{1,1,2,2}}
   {{1,2,2,2}}
   {{1,2,3,3}}
   {{1,2,3,4}}
   {{1,1},{1,1}}
   {{1,2},{1,2}}
   {{1,2},{2,2}}
   {{1,3},{2,3}}
   {{1},{1},{1},{1}}

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

Cf. A001055, A001970, A007716, A007718, A056156, A096827, A253249, A285573, A293994, A318099, A319557, A319616-A319646, A319721.

A096827 Number of antichains in divisor lattice D(n).

Original entry on oeis.org

2, 3, 3, 4, 3, 6, 3, 5, 4, 6, 3, 10, 3, 6, 6, 6, 3, 10, 3, 10, 6, 6, 3, 15, 4, 6, 5, 10, 3, 20, 3, 7, 6, 6, 6, 20, 3, 6, 6, 15, 3, 20, 3, 10, 10, 6, 3, 21, 4, 10, 6, 10, 3, 15, 6, 15, 6, 6, 3, 50, 3, 6, 10, 8, 6, 20, 3, 10, 6, 20, 3, 35, 3, 6, 10, 10, 6, 20, 3, 21, 6, 6, 3, 50, 6, 6, 6, 15, 3, 50, 6

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.
The empty set is counted as an antichain in D(n).
a(n) = gamma(n+1) where gamma is degree of cardinal completeness of Łukasiewicz n-valued logic. - Artur Jasinski, Mar 01 2010

Alexander S. Karpenko, Lukasiewicz's Logics and Prime Numbers, Luniver Press, Beckington, 2006. See Table I p. 113.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..990
Index entries for sequences related to Łukasiewicz

Cf. A096825, A096826, A097699.
Cf. A175177, A175178. - Artur Jasinski, Mar 01 2010
Cf. A000005, A008480, A253249, A285572 A285573.

nn=200;
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]]]];
Table[Length[stableSets[Divisors[n],Divisible]],{n,nn}] (* Gus Wiseman, Aug 24 2018 *)

a(n) = A285573(n) + 1. - Gus Wiseman, Aug 24 2018

More terms from John W. Layman, Aug 20 2004

A096825 Maximal size of an antichain in divisor lattice D(n).

Original entry on oeis.org

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

Offset: 1

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

nonn

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

Also the number of divisors of n with half (rounded either way) as many prime factors (counting multiplicity) as n. - Gus Wiseman, Aug 24 2018

			There are two maximal size antichains of divisors of 180, namely {12, 18, 20, 30, 45} and {4, 6, 9, 10, 15}. Both have length 5 so a(180) = 5. - _Gus Wiseman_, Aug 24 2018

Eric M. Schmidt, Table of n, a(n) for n = 1..10000
S.-H. Cha, E. G. DuCasse and L. V. Quintas, Graph invariants based on the divides relation and ordered by prime signatures, arXiv:1405.5283 [math.NT] (2014), (2.19).

Cf. A001055, A008480, A096826, A096827, A253249, A285573.

a:=proc(n) local klist,x; klist:=ifactors(n)[2,1..-1,2]; coeff(normal(mul((1-x^(k+1))/(1-x),k=klist)),x,floor(add(k,k=klist)/2)) end: seq(a(n), n=1..100);

a[n_] := Module[{pp, kk, x}, {pp, kk} = Transpose[FactorInteger[n]]; Coefficient[ Product[ Total[x^Range[0, k]], {k, kk}], x, Quotient[ Total[ kk], 2] ] ]; Array[a, 100] (* Jean-François Alcover, Nov 20 2017 *)
Table[Length[Select[Divisors[n],PrimeOmega[#]==Round[PrimeOmega[n]/2]&]],{n,50}] (* Gus Wiseman, Aug 24 2018 *)

a(n)=if(n<6||isprimepower(n), return(1)); my(d=divisors(n),r=1,u); d=d[2..#d-1];for(k=0,2^#d-1,if(hammingweight(k)<=r,next); u=vecextract(d,k); for(i=1,#u, for(j=i+1,#u, if(u[j]%u[i]==0, next(3))));r=#u);r \\ Charles R Greathouse IV, May 14 2013

from sympy import factorint
from sympy.utilities.iterables import multiset_combinations
def A096825(n):
    fs = factorint(n)
    return len(list(multiset_combinations(fs,sum(fs.values())//2))) # Chai Wah Wu, Aug 23 2021

def A096825(n) :
    if n==1 : return 1
    R. = QQ[]; mults = [x[1] for x in factor(n)]
    return prod((t^(m+1)-1)//(t-1) for m in mults)[sum(mults)//2]
# Eric M. Schmidt, May 11 2013

a(n) is the coefficient at x^k in (1+x+...+x^k_1)*...*(1+x+...+x^k_q) where n=p_1^k_1*...*p_q^k_q is the prime factorization of n and k=floor((k_1+...+k_q)/2). - Alec Mihailovs (alec(AT)mihailovs.com), Aug 22 2004

More terms from Alec Mihailovs (alec(AT)mihailovs.com), Aug 22 2004

A371128 Number of strict integer partitions of n containing all distinct divisors of all parts.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 1, 2, 1, 2, 2, 3, 3, 3, 5, 3, 5, 6, 7, 7, 8, 8, 9, 12, 13, 13, 14, 15, 16, 19, 23, 25, 26, 26, 27, 36, 37, 40, 42, 46, 50, 55, 66, 65, 71, 71, 82, 90, 102, 103, 114, 117, 130, 147, 154, 166, 176, 182, 194, 228, 239, 259, 267, 287, 307, 336

Offset: 0

Gus Wiseman, Mar 18 2024

nonn

Also strict integer partitions such that the number of parts is equal to the number of distinct divisors of all parts.

			The a(9) = 1 through a(19) = 7 partitions (A..H = 10..17):
  531  721   731   B1    751   D1    B31    D21    B51    H1     B71
       4321  5321  5421  931   B21   7521   7531   D31    9531   D51
                   6321  7321  7421  8421   64321  B321   A521   B521
                                     9321          65321  B421   D321
                                     54321         74321  75321  75421
                                                          84321  76321
                                                                 94321

The LHS is represented by A001221, distinct case of A001222.
The RHS is represented by A370820, for prime factors A303975.
Strict case of A371130 (ranks A370802) and A371178 (ranks A371177).
The complement is counted by A371180, non-strict A371132.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length.
A305148 counts partitions without divisors, strict A303362, ranks A316476.
Cf. A000837, A003963, A239312, A285573, A319055, A370803, A370808, A371171, A371172, A371173.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&SubsetQ[#,Union@@Divisors/@#]&]],{n,0,30}]

A371177 Positive integers whose prime indices include all distinct divisors of all prime indices.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 30, 32, 34, 36, 40, 42, 44, 48, 50, 54, 60, 62, 64, 66, 68, 72, 80, 82, 84, 88, 90, 96, 100, 102, 108, 110, 118, 120, 124, 126, 128, 132, 134, 136, 144, 150, 160, 162, 164, 166, 168, 170, 176, 180, 186, 192, 198, 200

Offset: 1

Gus Wiseman, Mar 18 2024

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.

Also positive integers with as many distinct prime factors (A001221) as distinct divisors of prime indices (A370820).

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   10: {1,3}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   22: {1,5}
   24: {1,1,1,2}
   30: {1,2,3}
   32: {1,1,1,1,1}
   34: {1,7}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   48: {1,1,1,1,2}

The LHS is A001221, distinct case of A001222.
The RHS is A370820, for prime factors A303975.
For bigomega on the LHS we have A370802, counted by A371130.
For divisors on the LHS we have A371165, counted by A371172.
Partitions of this type are counted by A371178, strict A371128.
The complement is A371179, counted by A371132.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length.
A305148 counts partitions without divisors, strict A303362, ranks A316476.
Cf. A000837, A003963, A239312, A285573, A355529, A370813, A371168.

Select[Range[100],PrimeNu[#]==Length[Union @@ Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]

A001221(a(n)) = A370820(a(n)).

A087086 Primitive sets of integers, each subset mapped onto a unique binary integer, values here shown in decimal.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 12, 16, 18, 20, 22, 24, 28, 32, 40, 48, 56, 64, 66, 68, 70, 72, 76, 80, 82, 84, 86, 88, 92, 96, 104, 112, 120, 128, 132, 144, 148, 160, 176, 192, 196, 208, 212, 224, 240, 256, 258, 264, 272, 274, 280, 288, 296, 304, 312, 320, 322, 328, 336, 338, 344

Offset: 0

Alan Sutcliffe (alansut(AT)ntlworld.com), Aug 14 2003

A primitive set of integers has no pair of elements one of which divides the other. Each element i in a subset contributes 2^(i-1) to the binary value for that subset. The integers missing from the sequence correspond to nonprimitive subsets.

			a(10)=22 since the 10th primitive set counting from 0 is {5,3,2}, which maps onto 10110 binary = 22 decimal.
From _Gus Wiseman_, Oct 31 2019: (Start)
The sequence of terms together with their binary expansions and binary indices begins:
   0:       0 ~ {}
   1:       1 ~ {1}
   2:      10 ~ {2}
   4:     100 ~ {3}
   6:     110 ~ {2,3}
   8:    1000 ~ {4}
  12:    1100 ~ {3,4}
  16:   10000 ~ {5}
  18:   10010 ~ {2,5}
  20:   10100 ~ {3,5}
  22:   10110 ~ {2,3,5}
  24:   11000 ~ {4,5}
  28:   11100 ~ {3,4,5}
(End)

Alan Sutcliffe, Divisors and Common Factors in Sets of Integers, awaiting publication

A051026 gives the number of primitive subsets of the integers 1 to n.
The version for prime indices (rather than binary indices) is A316476.
The relatively prime case is A328671.
Partitions with no consecutive divisible parts are A328171.
Compositions without consecutive divisible parts are A328460.
A ranking of antichains is A326704.
Cf. A000120, A048793, A070939, A285572, A285573, A303362, A304713, A305148, A328593.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[0,100],stableQ[Join@@Position[Reverse[IntegerDigits[#,2]],1],Divisible]&] (* Gus Wiseman, Oct 31 2019 *)

A304717 Number of connected strict integer partitions of n with pairwise indivisible parts.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 1, 3, 2, 4, 3, 5, 2, 5, 4, 6, 3, 7, 6, 9, 5, 9, 8, 13, 10, 15, 9, 15, 13, 18, 14, 22, 21, 26, 19, 29, 24, 36, 31, 40, 35, 45, 38, 54, 55, 59, 55, 70, 69, 84, 74, 89, 86, 107, 103, 119, 115, 143, 143, 159

Offset: 1

Gus Wiseman, May 17 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 multiset S is said to be connected if G(S) is a connected graph.

			The a(34) = 13 connected strict integer partitions with pairwise indivisible parts are (34), (18,16), (20,14), (22,12), (24,10), (26,8), (28,6), (30,4), (14,12,8), (15,10,9), (20,8,6), (14,10,6,4), (15,9,6,4). Their corresponding multiset multisystems (see A112798, A302242) are the following.
         (34): {{1,7}}
       (30 4): {{1,2,3},{1,1}}
       (28 6): {{1,1,4},{1,2}}
       (26 8): {{1,6},{1,1,1}}
      (24 10): {{1,1,1,2},{1,3}}
      (22 12): {{1,5},{1,1,2}}
      (20 14): {{1,1,3},{1,4}}
     (20 8 6): {{1,1,3},{1,1,1},{1,2}}
      (18 16): {{1,2,2},{1,1,1,1}}
    (15 10 9): {{2,3},{1,3},{2,2}}
   (15 9 6 4): {{2,3},{2,2},{1,2},{1,1}}
    (14 12 8): {{1,4},{1,1,2},{1,1,1}}
  (14 10 6 4): {{1,4},{1,3},{1,2},{1,1}}

Cf. A000009, A003963, A006126, A048143, A051424, A054921, A076078, A259936, A281116, A285572, A285573, A286518, A286520, A293993, A302242, A303362, A304714, A304716.

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[Select[IntegerPartitions[n],UnsameQ@@#&&Length[zsm[#]]===1&&Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]==={}&]],{n,30}]

A318726 Number of integer compositions of n that have only one part or whose consecutive parts are indivisible and the last and first part are also indivisible.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 5, 3, 8, 13, 12, 23, 27, 56, 64, 100, 150, 216, 325, 459, 700, 1007, 1493, 2186, 3203, 4735, 6929, 10243, 14952, 22024, 32366, 47558, 69906, 102634, 150984, 221713, 325919, 478842, 703648, 1034104, 1519432, 2233062, 3281004, 4821791, 7085359

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

			The a(10) = 13 compositions:
  (10)
  (7,3) (3,7) (6,4) (4,6)
  (5,3,2) (5,2,3) (3,5,2) (3,2,5) (2,5,3) (2,3,5)
  (3,2,3,2) (2,3,2,3)
The a(11) = 12 compositions:
  (11)
  (9,2) (2,9) (8,3) (3,8) (7,4) (4,7) (6,5) (5,6)
  (5,2,4) (4,5,2) (2,4,5)

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000740, A008965, A167606, A285573, A296302, A303362, A304713, A316476, A318727.

Table[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,({_,x_,y_,_}/;Divisible[x,y])|({y_,_,x_}/;Divisible[x,y])]&]//Length,{n,20}]

b(n,k,pred)={my(M=matrix(n,n)); for(n=1, n, M[n,n]=pred(k,n); for(j=1, n-1, M[n,j]=sum(i=1, n-j, if(pred(i,j), M[n-j,i], 0)))); sum(i=1, n, if(pred(i,k), M[n,i], 0))}
a(n)={1 + sum(k=1, n-1, b(n-k, k, (i,j)->i%j<>0))} \\ Andrew Howroyd, Sep 08 2018

a(n) = A328598(n) + 1. - Gus Wiseman, Nov 04 2019

a(21)-a(28) from Robert Price, Sep 08 2018
Terms a(29) and beyond from Andrew Howroyd, Sep 08 2018
Name corrected by Gus Wiseman, Nov 04 2019

A318729 Number of cyclic compositions (necklaces of positive integers) summing to n that have only one part or whose consecutive parts (including the last with first) are indivisible.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 4, 6, 6, 8, 11, 19, 21, 30, 41, 59, 79, 112, 157, 219, 305, 430, 605, 860, 1210, 1727, 2424, 3463, 4905, 7001, 9954, 14211, 20271, 28980, 41392, 59254, 84800, 121540, 174163, 249932, 358578, 515091, 739933, 1063827, 1529767, 2201383

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

			The a(13) = 11 cyclic compositions with successive parts indivisible:
  (13)
  (2,11) (3,10) (4,9) (5,8) (6,7)
  (2,4,7) (2,6,5) (2,8,3) (3,6,4)
  (2,3,5,3)

Andrew Howroyd, Table of n, a(n) for n = 1..100

Cf. A000740, A008965, A059966, A167606, A285573, A303362, A304713, A316476, A318726, A318727, A318728, A318730, A328600.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[Length[#]==1,neckQ[#]&&And@@Not/@Divisible@@@Partition[#,2,1,1]]&]],{n,20}]

b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q,]}
seq(n)={my(v=sum(k=1, n, k*b(n, k, (i,j)->i%j<>0))); vector(n, n, 1 + sumdiv(n, d, eulerphi(d)*v[n/d])/n)} \\ Andrew Howroyd, Oct 27 2019

a(n) = A328600(n) + 1. - Andrew Howroyd, Oct 27 2019

Terms a(21) and beyond from Andrew Howroyd, Sep 08 2018

Name corrected by Gus Wiseman, Nov 04 2019

cp's OEIS Frontend

A316475 Number of locally stable rooted trees with n nodes, meaning no branch is a submultiset of any other (unequal) branch of the same root.

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A319719 Number of non-isomorphic connected antichains of multisets of weight n.

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A096827 Number of antichains in divisor lattice D(n).

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A096825 Maximal size of an antichain in divisor lattice D(n).

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A371128 Number of strict integer partitions of n containing all distinct divisors of all parts.

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A371177 Positive integers whose prime indices include all distinct divisors of all prime indices.

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A087086 Primitive sets of integers, each subset mapped onto a unique binary integer, values here shown in decimal.

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Mathematica

A304717 Number of connected strict integer partitions of n with pairwise indivisible parts.

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