A096827 -id:A096827 - cp's OEIS frontend

A051026 Number of primitive subsequences of {1, 2, ..., n}.

1, 2, 3, 5, 7, 13, 17, 33, 45, 73, 103, 205, 253, 505, 733, 1133, 1529, 3057, 3897, 7793, 10241, 16513, 24593, 49185, 59265, 109297, 163369, 262489, 355729, 711457, 879937, 1759873, 2360641, 3908545, 5858113, 10534337, 12701537, 25403073, 38090337, 63299265, 81044097, 162088193, 205482593, 410965185, 570487233, 855676353

Offset: 0

Eric W. Weisstein

a(n) counts all subsequences of {1, ..., n} in which no term divides any other. If n is a prime a(n) = 2*a(n-1)-1 because for each subsequence s counted by a(n-1) two different subsequences are counted by a(n): s and s,n. There is only one exception: 1,n is not a primitive subsequence because 1 divides n. For all n>1: a(n) < 2*a(n-1). - Alois P. Heinz, Mar 07 2011

Maximal primitive subsets are counted by A326077. - Gus Wiseman, Jun 07 2019

			a(4) = 7, the primitive subsequences (including the empty sequence) are: (), (1), (2), (3), (4), (2,3), (3,4).
a(5) = 13 = 2*7-1, the primitive subsequences are: (), (5), (1), (2), (2,5), (3), (3,5), (4), (4,5), (2,3), (2,3,5), (3,4), (3,4,5).
From _Gus Wiseman_, Jun 07 2019: (Start)
The a(0) = 1 through a(5) = 13 primitive (pairwise indivisible) subsets:
  {}  {}   {}   {}     {}     {}
      {1}  {1}  {1}    {1}    {1}
           {2}  {2}    {2}    {2}
                {3}    {3}    {3}
                {2,3}  {4}    {4}
                       {2,3}  {5}
                       {3,4}  {2,3}
                              {2,5}
                              {3,4}
                              {3,5}
                              {4,5}
                              {2,3,5}
                              {3,4,5}
a(n) is also the number of subsets of {1..n} containing all of their pairwise products <= n as well as any quotients of divisible elements. For example, the a(0) = 1 through a(5) = 13 subsets are:
  {}  {}   {}     {}       {}         {}
      {1}  {1}    {1}      {1}        {1}
           {1,2}  {1,2}    {1,3}      {1,3}
                  {1,3}    {1,4}      {1,4}
                  {1,2,3}  {1,2,4}    {1,5}
                           {1,3,4}    {1,2,4}
                           {1,2,3,4}  {1,3,4}
                                      {1,3,5}
                                      {1,4,5}
                                      {1,2,3,4}
                                      {1,2,4,5}
                                      {1,3,4,5}
                                      {1,2,3,4,5}
Also the number of subsets of {1..n} containing all of their multiples <= n. For example, the a(0) = 1 through a(5) = 13 subsets are:
  {}  {}   {}     {}       {}         {}
      {1}  {2}    {2}      {3}        {3}
           {1,2}  {3}      {4}        {4}
                  {2,3}    {2,4}      {5}
                  {1,2,3}  {3,4}      {2,4}
                           {2,3,4}    {3,4}
                           {1,2,3,4}  {3,5}
                                      {4,5}
                                      {2,3,4}
                                      {2,4,5}
                                      {3,4,5}
                                      {2,3,4,5}
                                      {1,2,3,4,5}
(End)
From _Gus Wiseman_, Mar 12 2024: (Start)
Also the number of subsets of {1..n} containing all divisors of the elements. For example, the a(0) = 1 through a(6) = 17 subsets are:
  {}  {}   {}     {}       {}         {}
      {1}  {1}    {1}      {1}        {1}
           {1,2}  {1,2}    {1,2}      {1,2}
                  {1,3}    {1,3}      {1,3}
                  {1,2,3}  {1,2,3}    {1,5}
                           {1,2,4}    {1,2,3}
                           {1,2,3,4}  {1,2,4}
                                      {1,2,5}
                                      {1,3,5}
                                      {1,2,3,4}
                                      {1,2,3,5}
                                      {1,2,4,5}
                                      {1,2,3,4,5}
(End)

Blanchet-Sadri, Francine. Algorithmic combinatorics on partial words. Chapman & Hall/CRC, Boca Raton, FL, 2008. ii+385 pp. ISBN: 978-1-4200-6092-8; 1-4200-6092-9 MR2384993 (2009f:68142). See p. 320. - N. J. A. Sloane, Apr 06 2012

Juliana Couras, Ricardo Jesus, and Tomás Oliveira e Silva, Table of n, a(n) for n = 0..800 (terms up to n=80 from Alois P. Heinz)
Marcel K. Goh and Jonah Saks, Alternating-sum statistics for certain sets of integers, arXiv:2206.12535 [math.CO], 2022.
Nathan McNew, Counting primitive subsets and other statistics of the divisor graph of {1,2,..,n}, arXiv:1808.04923 [math.NT], 2018.
Richárd Palincza, Counting type and extremal problems from Arithmetic Combinatorics, Ph. D. Thesis, Budapest Univ. Tech. Econ. (Hungary, 2024).
Eric Weisstein's World of Mathematics, Primitive Sequence

Cf. A007865, A054519, A096827, A103580, A303362, A305148.
Cf. A326023, A326076, A326077, A326081, A326082, A326083, A326117.
Cf. A037031, A051026, A355740, A368110, A370585.

with(numtheory):
b:= proc(s) option remember; local n;
      n:= max(s[]);
      `if`(n<0, 1, b(s minus {n}) + b(s minus divisors(n)))
    end:
bb:= n-> b({$2..n} minus divisors(n)):
sb:= proc(n) option remember; `if`(n<2, 0, bb(n) + sb(n-1)) end:
a:= n-> `if`(n=0, 1, `if`(isprime(n), 2*a(n-1)-1, 2+sb(n))):
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 07 2011

b[s_] := b[s] = With[{n=Max[s]}, If[n < 0, 1, b[Complement[s, {n}]] + b[Complement[s, Divisors[n]]]]];
bb[n_] := b[Complement[Range[2, n], Divisors[n]]];
sb[n_] := sb[n] = If[n < 2, 0, bb[n] + sb[n-1]];
a[n_] := If[n == 0, 1, If[PrimeQ[n], 2a[n-1] - 1, 2 + sb[n]]]; Table[a[n], {n, 0, 37}]
(* Jean-François Alcover, Jul 27 2011, converted from Maple *)
Table[Length[Select[Subsets[Range[n]], SubsetQ[#,Select[Union@@Table[#*i,{i,n}],#<=n&]]&]],{n,10}] (* Gus Wiseman, Jun 07 2019 *)
Table[Length[Select[Subsets[Range[n]], #==Union@@Divisors/@#&]],{n,0,10}] (* Gus Wiseman, Mar 12 2024 *)

More terms from David Wasserman, May 02 2002

a(32)-a(37) from Donovan Johnson, Aug 11 2010

A319721 Number of non-isomorphic antichains of multisets of weight n.

Original entry on oeis.org

1, 1, 4, 8, 24, 50, 148, 349, 1014, 2717, 8114

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.

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 8 antichains:
1: {{1}}
2: {{1,1}}
   {{1,2}}
   {{1},{1}}
   {{1},{2}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1,2,3}}
   {{1},{2,2}}
   {{1},{2,3}}
   {{1},{1},{1}}
   {{1},{2},{2}}
   {{1},{2},{3}}

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

Cf. A001055, A001970, A007716, A096827, A253249, A285572, A285573, A293993, A318099, A319616-A319646, A319719.

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.

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

A326077 Number of maximal primitive subsets of {1..n}.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 4, 6, 7, 11, 11, 13, 13, 23, 24, 36, 36, 48, 48, 64, 66, 126, 126, 150, 151, 295, 363, 507, 507, 595, 595, 895, 903, 1787, 1788, 2076, 2076, 4132, 4148, 5396, 5396, 6644, 6644, 9740, 11172, 22300, 22300, 26140, 26141, 40733, 40773, 60333, 60333, 80781, 80783

Offset: 0

Gus Wiseman, Jun 05 2019

nonn

a(n) is the number of maximal primitive subsets of {1, ..., n}. Here primitive means that no element of the subset divides any other and maximal means that no element can be added to the subset while maintaining the property of being pairwise indivisible. - Nathan McNew, Aug 10 2020

			The a(0) = 1 through a(9) = 7 sets:
  {}  {1}  {1}  {1}   {1}   {1}    {1}    {1}     {1}     {1}
           {2}  {23}  {23}  {235}  {235}  {2357}  {2357}  {2357}
                      {34}  {345}  {345}  {3457}  {3457}  {2579}
                                   {456}  {4567}  {3578}  {3457}
                                                  {4567}  {3578}
                                                  {5678}  {45679}
                                                          {56789}

Nathan McNew, Table of n, a(n) for n = 0..300

Cf. A001055, A051026 (all primitive subsets), A067992, A096827, A143824, A285572, A285573, A303362, A305148, A305149, A316476, A325861, A326023, A326082.

stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
fasmax[y_]:=Complement[y, Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Range[n]],stableQ[#,Divisible]&]]],{n,0,10}]

divset(n)={sumdiv(n, d, if(dif(k>#p, ismax(b), my(f=!bitand(p[k], b)); if(!f || bittest(d, k), self()(k+1, b)) + if(f, self()(k+1, b+(1<Andrew Howroyd, Aug 30 2019

Terms a(19) to a(55) from Andrew Howroyd, Aug 30 2019

Name edited by Nathan McNew, Aug 10 2020

A321679 Number of non-isomorphic weight-n antichains (not necessarily strict) of sets.

Original entry on oeis.org

1, 1, 3, 5, 12, 19, 45, 75, 170, 314, 713

Offset: 0

Gus Wiseman, Nov 16 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(1) = 1 through a(5) = 19 antichains:
  {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}        {{1,2,3,4,5}}
         {{1},{1}}  {{1},{2,3}}    {{1,2},{1,2}}      {{1},{2,3,4,5}}
         {{1},{2}}  {{1},{1},{1}}  {{1},{2,3,4}}      {{1,2},{3,4,5}}
                    {{1},{2},{2}}  {{1,2},{3,4}}      {{1,4},{2,3,4}}
                    {{1},{2},{3}}  {{1,3},{2,3}}      {{1},{1},{2,3,4}}
                                   {{1},{1},{2,3}}    {{1},{2,3},{2,3}}
                                   {{1},{2},{3,4}}    {{1},{2},{3,4,5}}
                                   {{1},{1},{1},{1}}  {{1},{2,3},{4,5}}
                                   {{1},{1},{2},{2}}  {{1},{2,4},{3,4}}
                                   {{1},{2},{2},{2}}  {{1},{1},{1},{2,3}}
                                   {{1},{2},{3},{3}}  {{1},{2},{2},{3,4}}
                                   {{1},{2},{3},{4}}  {{1},{2},{3},{4,5}}
                                                      {{1},{1},{1},{1},{1}}
                                                      {{1},{1},{2},{2},{2}}
                                                      {{1},{2},{2},{2},{2}}
                                                      {{1},{2},{2},{3},{3}}
                                                      {{1},{2},{3},{3},{3}}
                                                      {{1},{2},{3},{4},{4}}
                                                      {{1},{2},{3},{4},{5}}

Cf. A006126, A049311, A096827, A285572, A293993, A293994, A318099, A319719, A319721, A320799, A321678.

A175177 Conjectured number of numbers for which the iteration x -> phi(x) + 1 terminates at prime(n). Cardinality of rooted tree T_p (where p is n-th prime) in Karpenko's book.

Original entry on oeis.org

2, 3, 4, 9, 2, 31, 6, 4, 2, 2, 2, 11, 24, 41, 2, 2, 2, 57, 2, 2, 58, 2, 2, 6, 17, 4, 2, 2, 39, 67, 2, 2, 2, 2, 2, 2, 25, 4, 2, 2, 2, 158, 2, 61, 2, 2, 2, 2, 2, 2, 54, 2, 186, 2, 10, 2, 2, 2, 18, 8, 2, 2, 2, 2, 96, 2, 2, 18, 2, 6, 15, 2, 2, 2, 2, 2, 2, 44, 34, 6, 2, 16, 2, 105, 2, 2, 60, 5, 4, 2, 2, 2, 4

Offset: 1

Artur Jasinski, Mar 01 2010

nonn

			a(3) = 4 because x = { 5, 8, 10, 12 } are the 4 numbers from which the iteration x -> phi(x) + 1 terminates at prime(3) = 5.
a(4) = 8 because x = { 7, 9, 14, 15, 16, 18, 20, 24, 30 } are the 9 numbers from which the iteration x -> phi(x) + 1 terminates at prime(4) = 7.

Richard K. Guy, Unsolved Problems in Number Theory, Third Edition, Springer, New York 2004. Chapter B41, Iterations of phi and sigma, page 148.
A. S. Karpenko, Lukasiewicz's Logics and Prime Numbers, (English translation), 2006. See Table 2 on p.125 ff.
A. S. Karpenko, Lukasiewicz's Logics and Prime Numbers, (Russian), 2000.

Hugo Pfoertner, Table of n, a(n) for n = 1..1000

Cf. A039649, A039650, A039651, A039652, A096827, A175178.

iterat(x) = {my(k,s); if ( isprime(x),return(x)); s=x;
for (k=1,1000000000,s=eulerphi(s)+1;if(isprime(s),return(s)));
return(s); }
check(y,endrange) = {my(count,start); count=0;
for(start=1,endrange,if(iterat(start)==y,count++;));
return(count); }
for (n=1,93,x=prime(n);print1(check(x,1000000),", "))
\\ Hugo Pfoertner, Sep 23 2017

Name clarified by Hugo Pfoertner, Sep 23 2017

A175178 a(n)=Values of cardinality of rooted trees CRT for successive primes.

Original entry on oeis.org

1, 1, 1, 2, 1, 5, 1, 1, 1, 1, 1, 2, 4, 6, 1, 1, 2, 9, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 5, 6, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 16, 1, 9, 2, 1, 1, 1, 1, 1, 7, 1, 19, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 6, 3, 1, 1, 2, 1, 11, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1

Offset: 1

Artur Jasinski, Mar 01 2010

nonn

Karpenko A.S. 2006. Lukasiewicz's Logics and Prime Numbers (English translation).
Karpenko A.S. 2000. Lukasiewicz's Logics and Prime Numbers (Russian).

A058340, A058340, A096827, A175177.

A318394 Number of finite sets of set partitions of {1,...,n} such that any two have meet {{1},...,{n}}.

Original entry on oeis.org

2, 4, 18, 316, 37492

Offset: 1

Gus Wiseman, Aug 25 2018

			The a(3) = 18 sets of set partitions:
        0
    {{1,2,3}}
   {{1,3},{2}}
   {{1,2},{3}}
   {{1},{2,3}}
  {{1},{2},{3}}
   {{1,2},{3}}   {{1,3},{2}}
   {{1},{2,3}}   {{1,3},{2}}
   {{1},{2,3}}   {{1,2},{3}}
  {{1},{2},{3}}   {{1,2,3}}
  {{1},{2},{3}}  {{1,3},{2}}
  {{1},{2},{3}}  {{1,2},{3}}
  {{1},{2},{3}}  {{1},{2,3}}
   {{1},{2,3}}   {{1,2},{3}}  {{1,3},{2}}
  {{1},{2},{3}}  {{1,2},{3}}  {{1,3},{2}}
  {{1},{2},{3}}  {{1},{2,3}}  {{1,3},{2}}
  {{1},{2},{3}}  {{1},{2,3}}  {{1,2},{3}}
  {{1},{2},{3}}  {{1},{2,3}}  {{1,2},{3}}  {{1,3},{2}}

Cf. A000110, A000258, A008277, A059849, A060639, A096827, A181939.

Cf. A318389, A318390, A318391, A318392, A318531, A318532, A318531, A318532.

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]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
spmeet[a_,b_]:=DeleteCases[Union@@Outer[Intersection,a,b,1],{}];spmeet[a_,b_,c__]:=spmeet[spmeet[a,b],c];
Table[Length[stableSets[sps[Range[n]],Max@@Length/@spmeet[#1,#2]>1&]],{n,5}]

cp's OEIS Frontend

A097699 Records in the numbers of antichains in the divisor lattice D(n) (A096827).

Views

Author

Keywords

Comments

Crossrefs

A051026 Number of primitive subsequences of {1, 2, ..., n}.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A319721 Number of non-isomorphic antichains of multisets of weight n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Sage

Formula

Extensions

A326077 Number of maximal primitive subsets of {1..n}.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A321679 Number of non-isomorphic weight-n antichains (not necessarily strict) of sets.

Views

Author

Keywords

Comments

Examples

Crossrefs

A175177 Conjectured number of numbers for which the iteration x -> phi(x) + 1 terminates at prime(n). Cardinality of rooted tree T_p (where p is n-th prime) in Karpenko's book.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs