A084422 -id:A084422 - cp's OEIS frontend

A324742 Number of subsets of {2...n} containing no prime indices of the elements.

1, 2, 3, 6, 10, 16, 24, 48, 84, 144, 228, 420, 648, 1080, 1800, 3600, 5760, 11136, 16704, 31104, 53568, 90624, 136896, 269952, 515712, 862080, 1708800, 3171840, 4832640, 9325440, 14890752, 29781504, 52245504, 88418304, 166017024, 331628544, 497645568, 829409280

Offset: 1

Gus Wiseman, Mar 15 2019

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 a(1) = 1 through a(6) = 16 subsets:
  {}  {}   {}   {}     {}       {}
      {2}  {2}  {2}    {2}      {2}
           {3}  {3}    {3}      {3}
                {4}    {4}      {4}
                {2,4}  {5}      {5}
                {3,4}  {2,4}    {6}
                       {2,5}    {2,4}
                       {3,4}    {2,5}
                       {4,5}    {3,4}
                       {2,4,5}  {3,6}
                                {4,5}
                                {4,6}
                                {5,6}
                                {2,4,5}
                                {3,4,6}
                                {4,5,6}
An example for n = 20 is {4,5,6,12,17,18,19}, with prime indices:
   4: {1,1}
   5: {3}
   6: {1,2}
  12: {1,1,2}
  17: {7}
  18: {1,2,2}
  19: {8}
None of these prime indices {1,2,3,7,8} belong to the set, as required.

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

The maximal case is A324763. The version for subsets of {1...n} is A324741. The strict integer partition version is A324752. The integer partition version is A324757. The Heinz number version is A324761. An infinite version is A304360.
Cf. A000720, A001462, A007097, A076078, A084422, A085945, A112798, A276625, A290689, A290822, A306844, A324764.
Cf. A324695, A324737, A324743, A324751, A324756, A324758.

Table[Length[Select[Subsets[Range[2,n]],Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]],{n,10}]

pset(n)={my(b=0,f=factor(n)[,1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n-1,k,pset(k+1)>>1), d=0); for(i=1, #p, d=bitor(d, p[i]));
((k,b)->if(k>#p, 1, my(t=self()(k+1,b)); if(!bitand(p[k], b), t+=if(bittest(d,k), self()(k+1, b+(1<Andrew Howroyd, Aug 16 2019

Terms a(21) and beyond from Andrew Howroyd, Aug 16 2019

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).

A225520 The number of subsets of the set of divisors of n in which elements are pairwise coprime.

Original entry on oeis.org

2, 4, 4, 6, 4, 10, 4, 8, 6, 10, 4, 16, 4, 10, 10, 10, 4, 16, 4, 16, 10, 10, 4, 22, 6, 10, 8, 16, 4, 30, 4, 12, 10, 10, 10, 26, 4, 10, 10, 22, 4, 30, 4, 16, 16, 10, 4, 28, 6, 16, 10, 16, 4, 22, 10, 22, 10, 10, 4, 50, 4, 10, 16, 14, 10, 30, 4, 16, 10, 30, 4, 36

Offset: 1

R. J. Mathar, May 09 2013

nonn

Note that this is not 1+A048691(n); n=30 is a counterexample.

The number of all subsets of the set of divisors (without the restriction) is 2^A000005(n), which therefore is an upper bound of the current sequence.

			For n=6, the set of divisors is {1,2,3,6} and the a(6)=10 subsets with pairwise coprime entries are {}, {1}, {2}, {3}, {6}, {1,2}, {1,3}, {1,6}, {2,3} and {1,2,3}.

T. D. Noe, Table of n, a(n) for n = 1..359

Cf. A076078 (subsets with lcm equal to n), A084422 (subsets of 1 through n).

paircoprime := proc(s)
    local L,i,j ;
    L := convert(s,list) ;
    for i from 1 to nops(L)-1 do
        for j from i+1 to nops(L) do
            if igcd(op(i,L),op(j,L)) <> 1 then
                return false;
            end if;
        end do:
    end do:
    return true;
end proc:
A225520 := proc(n)
    local dvs,a,p ;
    dvs := numtheory[divisors](n) ;
    a := 0 ;
    for p in combinat[powerset](dvs) do
        if paircoprime(p) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:

Table[Length[Select[Subsets[Divisors[n]], If[Length[#] < 2, True, If[Length[#] == 2, CoprimeQ @@ #, And @@ CoprimeQ @@ #]] &]], {n, 100}] (* T. D. Noe, May 09 2013 *)

A276187 Number of subsets of {1,..,n} of cardinality >= 2 such that the elements of each counted subset are pairwise coprime.

Original entry on oeis.org

0, 1, 4, 7, 18, 21, 48, 63, 94, 105, 220, 235, 482, 529, 600, 711, 1438, 1501, 3020, 3211, 3594, 3849, 7720, 7975, 11142, 11877, 14628, 15459, 30946, 31201, 62432, 69855, 76126, 80221, 89820, 91611, 183258, 192601, 208600, 214231, 428502, 431573, 863188, 900563

Offset: 1

Robert C. Lyons, Aug 23 2016

nonn

n is prime if and only if a(n) = 2*a(n-1)+n-1. - Robert Israel, Aug 24 2016

			From _Gus Wiseman_, May 08 2021: (Start)
The a(2) = 1 through a(6) = 21 sets:
  {1,2}   {1,2}    {1,2}     {1,2}      {1,2}
          {1,3}    {1,3}     {1,3}      {1,3}
          {2,3}    {1,4}     {1,4}      {1,4}
         {1,2,3}   {2,3}     {1,5}      {1,5}
                   {3,4}     {2,3}      {1,6}
                  {1,2,3}    {2,5}      {2,3}
                  {1,3,4}    {3,4}      {2,5}
                             {3,5}      {3,4}
                             {4,5}      {3,5}
                            {1,2,3}     {4,5}
                            {1,2,5}     {5,6}
                            {1,3,4}    {1,2,3}
                            {1,3,5}    {1,2,5}
                            {1,4,5}    {1,3,4}
                            {2,3,5}    {1,3,5}
                            {3,4,5}    {1,4,5}
                           {1,2,3,5}   {1,5,6}
                           {1,3,4,5}   {2,3,5}
                                       {3,4,5}
                                      {1,2,3,5}
                                      {1,3,4,5}
(End)

Robert Israel, Table of n, a(n) for n = 1..340

Cf. A000010, A002088, A018805.
The case of pairs is A015614.
The indivisible instead of coprime version is A051026(n) - n.
Allowing empty sets and singletons gives A084422.
The relatively prime instead of pairwise coprime version is A085945(n) - 1.
Allowing all singletons gives A187106.
Allowing only the singleton {1} gives A320426.
Row sums of A320436, each minus one.
The maximal case is counted by A343659.
The version for sets of divisors is A343655(n) - 1.
A000005 counts divisors.
A186972 counts pairwise coprime k-sets containing n.
A186974 counts pairwise coprime k-sets.
A326675 ranks pairwise coprime non-singleton sets.
Cf. A007360, A063647, A186971, A302696, A305713, A320423, A320430, A326077, A327516, A337485.

f:= proc(S) option remember;
    local s, Sp;
    if S = {} then return 1 fi;
    s:= S[-1];
    Sp:= S[1..-2];
    procname(Sp) + procname(select(t -> igcd(t,s)=1, Sp))
end proc:
seq(f({$1..n}) - n - 1, n=1..50); # Robert Israel, Aug 24 2016

f[S_] := f[S] = Module[{s, Sp}, If[S == {}, Return[1]]; s = S[[-1]]; Sp = S[[1;;-2]]; f[Sp] + f[Select[Sp, GCD[#, s] == 1&]]];
Table[f[Range[n]] - n - 1, {n, 1, 50}] (* Jean-François Alcover, Sep 15 2022, after Robert Israel *)

f(n,k=1)=if(n==1, return(2)); if(gcd(k,n)==1, f(n-1,n*k)) + f(n-1,k)
a(n)=f(n)-n-1 \\ Charles R Greathouse IV, Aug 24 2016

from sage.combinat.subsets_pairwise import PairwiseCompatibleSubsets
def is_coprime(x, y): return gcd(x, y) == 1
max_n = 40
seq = []
for n in range(1, max_n+1):
    P = PairwiseCompatibleSubsets(range(1,n+1), is_coprime)
    a_n = len([1 for s in P.list() if len(s) > 1])
    seq.append(a_n)
print(seq)

a(n) = A320426(n) - 1. - Gus Wiseman, May 08 2021

Name and example edited by Robert Israel, Aug 24 2016

A324763 Number of maximal subsets of {2...n} containing no prime indices of the elements.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 6, 6, 6, 6, 10, 10, 16, 16, 16, 16, 24, 24, 48, 48, 48, 48, 84, 84, 84, 84, 84, 84, 144, 144, 228, 228, 228, 228, 228, 228, 420, 420, 420, 420, 648, 648, 1080, 1080, 1080, 1080, 1800, 1800, 1800, 1800, 1800, 1800, 3600, 3600, 3600, 3600, 3600

Offset: 1

Gus Wiseman, Mar 17 2019

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 a(1) = 1 through a(9) = 6 subsets:
  {}  {2}  {2}  {2,4}  {3,4}    {2,4,5}  {2,4,5}  {2,4,5,8}  {2,4,5,8}
           {3}  {3,4}  {2,4,5}  {3,4,6}  {2,5,7}  {2,5,7,8}  {2,5,7,8}
                                {4,5,6}  {3,4,6}  {3,4,6,8}  {3,4,6,8,9}
                                         {3,6,7}  {3,6,7,8}  {3,6,7,8,9}
                                         {4,5,6}  {4,5,6,8}  {4,5,6,8,9}
                                         {5,6,7}  {5,6,7,8}  {5,6,7,8,9}

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

The non-maximal version is A324742.
The version for subsets of {1...n} is A324741.
An infinite version is A304360.
Cf. A076078, A084422, A085945, A112798, A276625, A306844, A324764.
Cf. A324695, A324743, A324751, A324756, A324758, A324762.

maxim[s_]:=Complement[s,Last/@Select[Tuples[s,2],UnsameQ@@#&&SubsetQ@@#&]];
Table[Length[maxim[Select[Subsets[Range[2,n]],Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]]],{n,10}]

pset(n)={my(b=0, f=factor(n)[, 1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n-1, k, pset(k+1)>>1), d=0); for(i=1, #p, d=bitor(d, p[i]));
my(ismax(b)=my(e=0); forstep(k=#p, 1, -1, if(bittest(b,k), e=bitor(e,p[k]), if(!bittest(e,k) && !bitand(p[k], b), return(0)) )); 1);
((k, b)->if(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 26 2019

Terms a(16) and beyond from Andrew Howroyd, Aug 26 2019

A320423 Number of set partitions of {1,...,n} where each block's elements are pairwise coprime.

Original entry on oeis.org

1, 1, 1, 2, 2, 8, 4, 28, 18, 120, 60, 888, 252, 5220, 1860, 22224, 9552, 311088, 59616, 2473056, 565920, 13627008, 4051872, 235039392, 33805440, 1932037632, 465239808, 20604487680, 4294865664, 386228795904, 35413136640

Offset: 0

Gus Wiseman, Jan 08 2019

Two or more numbers are pairwise coprime if no pair of them has a common divisor > 1. A single number is not considered to be pairwise coprime unless it is equal to 1.

			The a(5) = 8 set partitions:
  {{1},{2,3},{4,5}}
  {{1},{2,5},{3,4}}
   {{1,2},{3,4,5}}
   {{1,4},{2,3,5}}
   {{1,2,3},{4,5}}
   {{1,2,5},{3,4}}
   {{1,3,4},{2,5}}
   {{1,4,5},{2,3}}

Cf. A000110, A051424, A084422, A085945, A186974, A187106, A302569, A302696, A303139, A303140, A320424, A320426, A320430, A320768.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
Table[Length[spsu[Select[Subsets[Range[n]],CoprimeQ@@#&],Range[n]]],{n,10}]

a(17)-a(18) from Alois P. Heinz, Jan 17 2019

a(19)-a(30) from Christian Sievers, Nov 28 2024

A324737 Number of subsets of {2...n} containing every element of {2...n} whose prime indices all belong to the subset.

Original entry on oeis.org

1, 2, 3, 6, 8, 16, 24, 48, 84, 168, 216, 432, 648, 1296, 2448, 4896, 6528, 13056, 19584, 39168, 77760, 155520, 229248, 458496, 790272, 1580544, 3128832, 6257664, 9386496, 18772992, 24081408, 48162816, 95938560, 191877120, 378335232, 756670464, 1135005696, 2270011392

Offset: 1

Gus Wiseman, Mar 13 2019

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 the number of subsets of {2...n} with complement containing no term whose prime indices all belong to the subset.

			The a(1) = 1 through a(6) = 16 subsets:
  {}  {}   {}     {}       {}         {}
      {2}  {3}    {3}      {4}        {4}
           {2,3}  {4}      {5}        {5}
                  {2,3}    {3,5}      {6}
                  {3,4}    {4,5}      {3,5}
                  {2,3,4}  {2,3,5}    {4,5}
                           {3,4,5}    {4,6}
                           {2,3,4,5}  {5,6}
                                      {2,3,5}
                                      {3,4,5}
                                      {3,5,6}
                                      {4,5,6}
                                      {2,3,4,5}
                                      {2,3,5,6}
                                      {3,4,5,6}
                                      {2,3,4,5,6}
An example for n = 15 is {2, 3, 5, 8, 9, 10, 11, 15}. The numbers from 2 to 15 with all prime indices in the subset are {3, 5, 9, 11, 15}, which all belong to the subset, as required.

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

Cf. A000720, A001221, A001462, A007097, A084422, A085945, A112798, A276625, A290689, A290822, A304360, A306844.

Cf. A324697, A324698, A324736, A324738, A324748, A324753, A324755.

Table[Length[Select[Subsets[Range[2,n]],Function[set,SubsetQ[set,Select[Range[2,n],SubsetQ[set,PrimePi/@First/@FactorInteger[#]]&]]]]],{n,10}]

pset(n)={my(b=0, f=factor(n)[, 1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n-1, k, pset(k+1)>>1), d=0); for(i=1, #p, d=bitor(d, p[i]));
((k, b)->if(k>#p, 1, my(t=self()(k+1, b+(1<Andrew Howroyd, Aug 24 2019

Terms a(21) and beyond from Andrew Howroyd, Aug 24 2019

A324762 Number of maximal subsets of {2...n} containing no element whose prime indices all belong to the subset.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 4, 4, 6, 6, 8, 8, 16, 16, 16, 16, 16, 16, 32, 32, 40, 40, 52, 52, 64, 64, 72, 72, 144, 144, 176, 176, 200, 200, 232, 232, 464, 464, 464, 464, 536, 536, 1072, 1072, 1072, 1072, 2144, 2144, 2400, 2400, 2400, 2400, 4800, 4800, 4800, 4800, 4800

Offset: 1

Gus Wiseman, Mar 17 2019

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 a(2) = 1 through a(9) = 6 maximal subsets:
  {2}  {2}  {2,4}  {3,4}    {3,4,6}    {3,4,6}    {3,4,6,8}    {2,4,5,6,8}
       {3}  {3,4}  {2,4,5}  {2,4,5,6}  {3,6,7}    {3,6,7,8}    {2,5,6,7,8}
                                       {2,4,5,6}  {2,4,5,6,8}  {3,4,6,8,9}
                                       {2,5,6,7}  {2,5,6,7,8}  {3,6,7,8,9}
                                                               {4,5,6,8,9}
                                                               {5,6,7,8,9}

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

The non-maximal version is A324739.
The version for subsets of {1...n} is A324744.
An infinite version is A324694.
Cf. A076078, A084422, A085945, A112798, A276625, A304360, A306844.
Cf. A324695, A324738, A324742, A324743, A324763, A324750, A324754, A324755, A324760.

maxim[s_]:=Complement[s,Last/@Select[Tuples[s,2],UnsameQ@@#&&SubsetQ@@#&]];
Table[Length[maxim[Select[Subsets[Range[2,n]],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]]],{n,10}]

pset(n)={my(b=0, f=factor(n)[, 1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n, k, pset(k)), d=0); for(i=1, #p, d=bitor(d, p[i]));
my(ismax(b)=for(k=1, #p, if(!bittest(b,k) && bitnegimply(p[k], b), my(e=bitor(b, 1<#p, ismax(b), my(f=bitnegimply(p[k], b)); if(!f || bittest(d, k), self()(k+1, b)) + if(f, self()(k+1, b+(1<Andrew Howroyd, Aug 27 2019

Terms a(16) and beyond from Andrew Howroyd, Aug 27 2019

A343655 Number of pairwise coprime sets of divisors of n, where a singleton is not considered pairwise coprime unless it is {1}.

Original entry on oeis.org

1, 2, 2, 3, 2, 6, 2, 4, 3, 6, 2, 10, 2, 6, 6, 5, 2, 10, 2, 10, 6, 6, 2, 14, 3, 6, 4, 10, 2, 22, 2, 6, 6, 6, 6, 17, 2, 6, 6, 14, 2, 22, 2, 10, 10, 6, 2, 18, 3, 10, 6, 10, 2, 14, 6, 14, 6, 6, 2, 38, 2, 6, 10, 7, 6, 22, 2, 10, 6, 22, 2, 24, 2, 6, 10, 10, 6, 22, 2

Offset: 1

Gus Wiseman, Apr 26 2021

nonn

First differs from A015995 at a(210) = 88, A015995(210) = 86.

			For example, the a(n) subsets for n = 1, 2, 4, 6, 8, 12, 16, 24 are:
  {1}  {1}    {1}    {1}      {1}    {1}      {1}     {1}
       {1,2}  {1,2}  {1,2}    {1,2}  {1,2}    {1,2}   {1,2}
              {1,4}  {1,3}    {1,4}  {1,3}    {1,4}   {1,3}
                     {1,6}    {1,8}  {1,4}    {1,8}   {1,4}
                     {2,3}           {1,6}    {1,16}  {1,6}
                     {1,2,3}         {2,3}            {1,8}
                                     {3,4}            {2,3}
                                     {1,12}           {3,4}
                                     {1,2,3}          {3,8}
                                     {1,3,4}          {1,12}
                                                      {1,24}
                                                      {1,2,3}
                                                      {1,3,4}
                                                      {1,3,8}

The case of pairs is A063647.
The case of triples is A066620.
The version with empty sets and singletons is A225520.
A version for prime indices is A304711.
The version for strict integer partitions is A305713.
The version for subsets of {1..n} is A320426 = A276187 + 1.
The version for binary indices is A326675.
The version for integer partitions is A327516.
The version for standard compositions is A333227.
The maximal case is A343652.
The case without 1's is A343653.
The case without 1's with singletons is A343654.
The maximal case without 1's 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.
A325683 counts maximal Golomb rulers.
A326077 counts maximal pairwise indivisible sets.
Cf. A007360, A062319, A067824, A076078, A084422, A187106, A282935, A285572, A304709, A320423, A337485, A343659.

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

A066620 Number of unordered triples of distinct pairwise coprime divisors 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, 1, 3, 0, 13, 1, 2, 1, 1, 1, 5, 0, 2, 2, 4, 0, 7, 0

Offset: 1

K. B. Subramaniam (kb_subramaniambalu(AT)yahoo.com) and Amarnath Murthy, Dec 24 2001

nonn

a(m) = a(n) if m and n have same factorization structure.

			a(24) = 3: the divisors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. The triples are (1, 2, 3), (1, 2, 9), (1, 3, 4).
a(30) = 7: the triples are (1, 2, 3), (1, 2, 5), (1, 3, 5), (2, 3, 5), (1, 3, 10), (1, 5, 6), (1, 2, 15).

Amarnath Murthy, Decomposition of the divisors of a natural number into pairwise coprime sets, Smarandache Notions Journal, vol. 12, No. 1-2-3, Spring 2001.pp 303-306.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Positions of zeros are A000961.
Positions of ones are A006881.
The version for subsets of {1..n} instead of divisors is A015617.
The non-strict ordered version is A048785.
The version for pairs of divisors is A063647.
The non-strict version (3-multisets) is A100565.
The version for partitions is A220377 (non-strict: A307719).
A version for sets of divisors of any size is A225520.
A000005 counts divisors.
A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
A007304 ranks 3-part strict partitions.
A014311 ranks 3-part compositions.
A014612 ranks 3-part partitions.
A018892 counts unordered pairs of coprime divisors (ordered: A048691).
A051026 counts pairwise indivisible subsets of {1..n}.
A337461 counts 3-part pairwise coprime compositions.
A338331 lists Heinz numbers of pairwise coprime partitions.
Cf. A007360, A023022, A084422, A276187, A282935, A305713, A337563, A337605, A343652, A343655.

Table[Length[Select[Subsets[Divisors[n],{3}],CoprimeQ@@#&]],{n,100}] (* Gus Wiseman, Apr 28 2021 *)

A066620(n) = (numdiv(n^3)-3*numdiv(n)+2)/6; \\ After Jovovic's formula. - Antti Karttunen, May 27 2017

from sympy import divisor_count as d
def a(n): return (d(n**3) - 3*d(n) + 2)/6 # Indranil Ghosh, May 27 2017

In the reference it is shown that if k is a squarefree number with r prime factors and m with (r+1) prime factors then a(m) = 4*a(k) + 2^k - 1.

a(n) = (tau(n^3)-3*tau(n)+2)/6. - Vladeta Jovovic, Nov 27 2004

More terms from Vladeta Jovovic, Apr 03 2003
Name corrected by Andrey Zabolotskiy, Dec 09 2020
Name corrected by Gus Wiseman, Apr 28 2021 (ordered version is 6*a(n))

cp's OEIS Frontend

A324742 Number of subsets of {2...n} containing no prime indices of the elements.

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

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A225520 The number of subsets of the set of divisors of n in which elements are pairwise coprime.

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A276187 Number of subsets of {1,..,n} of cardinality >= 2 such that the elements of each counted subset are pairwise coprime.

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A324763 Number of maximal subsets of {2...n} containing no prime indices of the elements.

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A320423 Number of set partitions of {1,...,n} where each block's elements are pairwise coprime.

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A324737 Number of subsets of {2...n} containing every element of {2...n} whose prime indices all belong to the subset.

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