A326439 Number of maximal subsets of {1..n} such that no two elements have the same sorted prime signature.

1, 1, 1, 2, 2, 3, 3, 4, 4, 8, 16, 20, 20, 24, 36, 48, 48, 56, 112, 128, 192, 240, 288, 324, 324, 486, 567, 1134, 1512, 1680, 1680, 1848, 1848, 2112, 2376, 2640, 2640, 2880, 3168, 3456, 6912, 7488, 14976, 16128, 20160, 24192, 26208, 28080, 28080, 37440, 43680

Offset: 0

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(0) = 1 through a(9) = 8 subsets:
  {}  {1}  {12}  {12}  {124}  {124}  {1246}  {1246}  {12468}  {12468}
                 {13}  {134}  {134}  {1346}  {1346}  {13468}  {12689}
                              {145}  {1456}  {1456}  {14568}  {13468}
                                             {1467}  {14678}  {13689}
                                                              {14568}
                                                              {14678}
                                                              {15689}
                                                              {16789}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A001221, A001222, A025487, A064839, A085089, A112798, A118914, A124010, A181819, A324762, A325263, A325365, A326438, A326441.

Programs

Mathematica

prisig[n_]:=If[n==1,{},Sort[Last/@FactorInteger[n]]];
Table[Times@@(Length/@Split[Sort[Array[prisig,n]]]),{n,0,30}]

PARI

a(n)={if(n==0, 1, my(M=Map()); for(i=1, n, my(f=factor(i)[,2], s=sum(k=1, #f, x^f[k]), z); mapput(M, s, if(mapisdefined(M, s, &z), z + 1, 1))); vecprod(Mat(M)[,2]))} \\ Andrew Howroyd, Aug 30 2019