A324837 -id:A324837 - cp's OEIS frontend

A303838 Number of z-forests with least common multiple n > 1.

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 8, 1, 1, 2, 2, 2, 5, 1, 2, 2, 4, 1, 8, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 16, 1, 2, 3, 1, 2, 8, 1, 3, 2, 8, 1, 7, 1, 2, 3, 3, 2, 8, 1, 5, 1, 2, 1, 16, 2, 2

Offset: 1

Gus Wiseman, May 19 2018

nonn

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. A z-forest is a finite set of pairwise indivisible positive integers greater than 1 such that all connected components are z-trees, meaning they have clutter density -1.
This is a generalization to multiset systems of the usual definition of hyperforest (viz. hypergraph F such that two distinct hyperedges of F intersect in at most a common vertex and such that every cycle of F is contained in a hyperedge).
If n is squarefree with k prime factors, then a(n) = A134954(k).
Differs from A324837 at positions {1, 180, 210, ...}. For example, a(210) = 55, A324837(210) = 49.

			The a(60) = 16 z-forests together with the corresponding multiset systems (see A112798, A302242) are the following.
       (60): {{1,1,2,3}}
     (3,20): {{2},{1,1,3}}
     (4,15): {{1,1},{2,3}}
     (4,30): {{1,1},{1,2,3}}
     (5,12): {{3},{1,1,2}}
     (6,20): {{1,2},{1,1,3}}
    (10,12): {{1,3},{1,1,2}}
    (12,15): {{1,1,2},{2,3}}
    (12,20): {{1,1,2},{1,1,3}}
    (15,20): {{2,3},{1,1,3}}
    (3,4,5): {{2},{1,1},{3}}
   (3,4,10): {{2},{1,1},{1,3}}
    (4,5,6): {{1,1},{3},{1,2}}
   (4,6,10): {{1,1},{1,2},{1,3}}
   (4,6,15): {{1,1},{1,2},{2,3}}
  (4,10,15): {{1,1},{1,3},{2,3}}

Gus Wiseman, Table of n, a(n) for n = 1..250
Roland Bacher, On the enumeration of labelled hypertrees and of labelled bipartite trees, arXiv:1102.2708 [math.CO], 2011.

Cf. A006126, A030019, A048143, A076078, A112798, A134954, A275307, A285572, A286518, A286520, A293993, A293994, A302242, A303837, A304118.

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];
Table[Length[Select[Rest[Subsets[Rest[Divisors[n]]]],Function[s,LCM@@s==n&&And@@Table[zensity[Select[s,Divisible[m,#]&]]==-1,{m,zsm[s]}]&&Select[Tuples[s,2],UnsameQ@@#&&Divisible@@#&]=={}]]],{n,100}]

A333226 Least common multiple of the n-th composition in standard order.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 1, 4, 3, 2, 2, 3, 2, 2, 1, 5, 4, 6, 3, 6, 2, 2, 2, 4, 3, 2, 2, 3, 2, 2, 1, 6, 5, 4, 4, 3, 6, 6, 3, 4, 6, 2, 2, 6, 2, 2, 2, 5, 4, 6, 3, 6, 2, 2, 2, 4, 3, 2, 2, 3, 2, 2, 1, 7, 6, 10, 5, 12, 4, 4, 4, 12, 3, 6, 6, 3, 6, 6, 3, 10, 4, 6, 6, 6, 2, 2

Offset: 1

Gus Wiseman, Mar 26 2020

nonn

The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again.

The version for binary indices is A271410.
The version for prime indices is A290103.
Positions of first appearances are A333225.
Let q(k) be the k-th composition in standard order:
- The terms of q(k) are row k of A066099.
- The sum of q(k) is A070939(k).
- The product of q(k) is A124758(k).
- The GCD of q(k) is A326674(k).
- The LCM of q(k) is A333226(k).
Cf. A000120, A029931, A048793, A074971, A076078, A233564, A285572, A289508, A289509, A324837, A333227, A333492.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[LCM@@stc[n],{n,100}]

A343652 Number of maximal pairwise coprime sets of divisors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 3, 4, 1, 5, 1, 5, 2, 2, 2, 8, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 8, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 10, 1, 2, 4, 6, 2, 5, 1, 4, 2, 5, 1, 12, 1, 2, 4, 4, 2, 5, 1, 8, 4, 2, 1, 10, 2, 2

Offset: 1

Gus Wiseman, Apr 25 2021

nonn

Also the number of maximal pairwise coprime sets of divisors > 1 of n. For example, the a(n) sets for n = 12, 30, 36, 60, 120 are:
  {6}    {30}     {6}    {30}     {30}
  {12}   {2,15}   {12}   {60}     {60}
  {2,3}  {3,10}   {18}   {2,15}   {120}
  {3,4}  {5,6}    {36}   {3,10}   {2,15}
         {2,3,5}  {2,3}  {3,20}   {3,10}
                  {2,9}  {4,15}   {3,20}
                  {3,4}  {5,6}    {3,40}
                  {4,9}  {5,12}   {4,15}
                         {2,3,5}  {5,6}
                         {3,4,5}  {5,12}
                                  {5,24}
                                  {8,15}
                                  {2,3,5}
                                  {3,4,5}
                                  {3,5,8}

			The a(n) sets for n = 12, 30, 36, 60, 120:
  {1,6}    {1,30}     {1,6}    {1,30}     {1,30}
  {1,12}   {1,2,15}   {1,12}   {1,60}     {1,60}
  {1,2,3}  {1,3,10}   {1,18}   {1,2,15}   {1,120}
  {1,3,4}  {1,5,6}    {1,36}   {1,3,10}   {1,2,15}
           {1,2,3,5}  {1,2,3}  {1,3,20}   {1,3,10}
                      {1,2,9}  {1,4,15}   {1,3,20}
                      {1,3,4}  {1,5,6}    {1,3,40}
                      {1,4,9}  {1,5,12}   {1,4,15}
                               {1,2,3,5}  {1,5,6}
                               {1,3,4,5}  {1,5,12}
                                          {1,5,24}
                                          {1,8,15}
                                          {1,2,3,5}
                                          {1,3,4,5}
                                          {1,3,5,8}

The case of pairs is A063647.
The case of triples is A066620.
The non-maximal version counting empty sets and singletons is A225520.
The non-maximal version with no 1's is A343653.
The non-maximal version is A343655.
The version for subsets of {1..n} is A343659.
The case without 1's or singletons is A343660.
A018892 counts pairwise coprime unordered pairs of divisors.
A048691 counts pairwise coprime ordered pairs of divisors.
A048785 counts pairwise coprime ordered triples of divisors.
A084422, A187106, A276187, and A320426 count pairwise coprime sets.
A100565 counts pairwise coprime unordered triples of divisors.
A305713 counts pairwise coprime non-singleton strict partitions.
A324837 counts minimal subsets of {1...n} with least common multiple n.
A325683 counts maximal Golomb rulers.
A326077 counts maximal pairwise indivisible sets.
Cf. A005361, A007359, A051026, A062319, A067824, A074206, A146291, A285572, A325859, A326359, A326496, A326675, A343654.

fasmax[y_]:=Complement[y,Union@@Most@*Subsets/@y];
Table[Length[fasmax[Select[Subsets[Divisors[n]],CoprimeQ@@#&]]],{n,100}]

a(n) = A343660(n) + A005361(n).

A343659 Number of maximal pairwise coprime subsets of {1..n}.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 7, 9, 9, 10, 10, 12, 16, 19, 19, 20, 20, 22, 28, 32, 32, 33, 54, 61, 77, 84, 84, 85, 85, 94, 112, 123, 158, 161, 161, 176, 206, 212, 212, 214, 214, 229, 241, 260, 260, 263, 417, 428, 490, 521, 521, 526, 655, 674, 764, 818, 818, 820, 820, 874, 918, 975, 1182, 1189, 1189

Offset: 1

Gus Wiseman, Apr 26 2021

nonn

For this sequence, it does not matter whether singletons are considered pairwise coprime.
For n > 2, also the number of maximal pairwise coprime subsets of {2..n}.
For each prime p <= n, p divides exactly one element of each maximal subset. - Bert Dobbelaere, May 04 2021

			The a(1) = 1 through a(9) = 7 subsets:
  {1}  {12}  {123}  {123}  {1235}  {156}   {1567}   {1567}   {1567}
                    {134}  {1345}  {1235}  {12357}  {12357}  {12357}
                                   {1345}  {13457}  {13457}  {12579}
                                                    {13578}  {13457}
                                                             {13578}
                                                             {14579}
                                                             {15789}

Bert Dobbelaere, Table of n, a(n) for n = 1..500
Bert Dobbelaere, Python program

The case of pairs is A015614.
The case of triples is A015617.
The non-maximal version counting empty sets and singletons is A084422.
The non-maximal version counting singletons is A187106.
The non-maximal version is A320426(n) = A276187(n) + 1.
The version for indivisibility instead of coprimality is A326077.
The version for sets of divisors is A343652.
The version for sets of divisors > 1 is A343660.
A018892 counts coprime unordered pairs of divisors.
A051026 counts pairwise indivisible subsets of {1..n}.
A100565 counts pairwise coprime unordered triples of divisors.
Cf. A007360, A067824, A087087, A225520, A324837, A325683, A325859, A326358, A326496, A326675, A333227, A343653, A343655.

fasmax[y_]:=Complement[y,Union@@Most@*Subsets/@y];
Table[Length[fasmax[Select[Subsets[Range[n]],CoprimeQ@@#&]]],{n,15}]

More terms from Bert Dobbelaere, May 04 2021

A343653 Number of non-singleton pairwise coprime nonempty sets of divisors > 1 of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 7, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 7, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 13, 0, 1, 2, 0, 1, 7, 0, 2, 1, 7, 0, 6, 0, 1, 2, 2, 1, 7, 0, 4, 0, 1, 0, 13, 1, 1

Offset: 1

Gus Wiseman, Apr 25 2021

nonn

First differs from A066620 at a(210) = 36, A066620(210) = 35.

			The a(n) sets for n = 6, 12, 24, 30, 36, 60, 72, 96:
  {2,3}  {2,3}  {2,3}  {2,3}    {2,3}  {2,3}    {2,3}  {2,3}
         {3,4}  {3,4}  {2,5}    {2,9}  {2,5}    {2,9}  {3,4}
                {3,8}  {3,5}    {3,4}  {3,4}    {3,4}  {3,8}
                       {5,6}    {4,9}  {3,5}    {3,8}  {3,16}
                       {2,15}          {4,5}    {4,9}  {3,32}
                       {3,10}          {5,6}    {8,9}
                       {2,3,5}         {2,15}
                                       {3,10}
                                       {3,20}
                                       {4,15}
                                       {5,12}
                                       {2,3,5}
                                       {3,4,5}

The case of pairs is A089233.
The version with 1's, empty sets, and singletons is A225520.
The version for subsets of {1..n} is A320426.
The version for strict partitions is A337485.
The version for compositions is A337697.
The version for prime indices is A337984.
The maximal case with 1's is A343652.
The version with empty sets is a(n) + 1.
The version with singletons is A343654(n) - 1.
The version with empty sets and singletons is A343654.
The version with 1's is A343655.
The maximal case is A343660.
A018892 counts pairwise coprime unordered pairs of divisors.
A048691 counts pairwise coprime ordered pairs of divisors.
A048785 counts pairwise coprime ordered triples of divisors.
A051026 counts pairwise indivisible subsets of {1..n}.
A100565 counts pairwise coprime unordered triples of divisors.
A305713 counts pairwise coprime non-singleton strict partitions.
A343659 counts maximal pairwise coprime subsets of {1..n}.
Cf. A007359, A067824, A074206, A076078, A084422, A187106, A285572, A324837, A326675, A327516, A338315.

Table[Length[Select[Subsets[Rest[Divisors[n]]],CoprimeQ@@#&]],{n,100}]

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

A330225 Position of first appearance of n in A290103 = LCM of prime indices.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271

Offset: 1

Gus Wiseman, Mar 26 2020

nonn

Appears to be the prime numbers (A000040) with 2 replaced by 1 and 37 replaced by 35.

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 version for product instead of lcm is A318871
The version for standard compositions is A333225.
The version for binary indices is A333492.
Let q(k) be the prime indices of k:
- The product of q(k) is A003963(k).
- The sum of q(k) is A056239(k).
- The terms of q(k) are row k of A112798.
- The GCD of q(k) is A289508(k).
- The LCM of q(k) is A290103(k).
- The LCM of q(k) + 1 is A328219(k).
Cf. A000837, A074761, A074971, A076078, A285572, A289509, A290104, A319333, A324837, A328451, A331579, A333226.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
q=Table[If[n==1,1,LCM@@primeMS[n]],{n,100}];
Table[Position[q,i][[1,1]],{i,First[Split[Union[q],#1+1==#2&]]}]

cp's OEIS Frontend

A303838 Number of z-forests with least common multiple n > 1.

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A333226 Least common multiple of the n-th composition in standard order.

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A343652 Number of maximal pairwise coprime sets of divisors of n.

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A343659 Number of maximal pairwise coprime subsets of {1..n}.

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A343653 Number of non-singleton pairwise coprime nonempty sets of divisors > 1 of n.

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

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A330225 Position of first appearance of n in A290103 = LCM of prime indices.

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