A325263 Number of subsets of {1..n} containing n such that no two elements have the same sorted prime signature.

1, 2, 2, 6, 4, 16, 8, 40, 40, 60, 36, 216, 72, 168, 168, 840, 240, 960, 360, 1080, 864, 864, 672, 6720, 4480, 2560, 10240, 7680, 3840, 42240, 7680, 92160, 23040, 23040, 23040, 253440, 42240, 49920, 49920, 324480, 74880, 524160, 112320, 336960, 336960, 181440

Offset: 1

Author

Gus Wiseman, Jul 06 2019

Keywords

• nonn

Comments

The sorted prime signature (A118914) of a positive integer is the multiset of exponents in its standard factorization into prime numbers.

Examples

			The a(1) = 1 through a(5) = 16 subsets:
  {1}  {2}    {3}    {4}      {5}      {6}        {7}
       {1,2}  {1,3}  {1,4}    {1,5}    {1,6}      {1,7}
                     {2,4}    {4,5}    {2,6}      {4,7}
                     {3,4}    {1,4,5}  {3,6}      {6,7}
                     {1,2,4}           {4,6}      {1,4,7}
                     {1,3,4}           {5,6}      {1,6,7}
                                       {1,2,6}    {4,6,7}
                                       {1,3,6}    {1,4,6,7}
                                       {1,4,6}
                                       {1,5,6}
                                       {2,4,6}
                                       {3,4,6}
                                       {4,5,6}
                                       {1,2,4,6}
                                       {1,3,4,6}
                                       {1,4,5,6}
		

Crossrefs

Cf. A001221, A001222, A025487, A064839, A085089, A112798, A118914, A124010, A181819, A325365, A326438, A326439, A326441.

Programs

Mathematica

prisig[n_]:=If[n==1,{},Sort[Last/@FactorInteger[n]]];
Table[Times@@(1+Length/@Split[Sort[Array[prisig,n]]])/(1+Count[Array[prisig,n],prisig[n]]),{n,30}]

Formula

a(n) = A326438(n)/(1 + A064839(n)).