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

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

Author

Gus Wiseman, Apr 25 2021

Keywords

• nonn

Comments

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}

Examples

			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}
		

Crossrefs

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.

Programs

Mathematica

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

Formula

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