A325365 Number of maximal subsets of {1..n} containing n such that no two elements have the same sorted prime signature.

1, 1, 1, 2, 1, 3, 1, 4, 4, 8, 4, 20, 4, 12, 12, 48, 8, 56, 16, 64, 48, 48, 36, 324, 162, 81, 567, 378, 168, 1680, 168, 1848, 264, 264, 264, 2640, 240, 288, 288, 3456, 576, 7488, 1152, 4032, 4032, 2016, 1872, 28080, 9360, 6240, 3360, 6720, 3584, 28672, 6144

Offset

1,4

Comments

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

Links

Table of n, a(n) for n=1..55.

Formula

a(n) = A326439(n)/A064839(n).

Example

The a(1) = 1 through a(12) = 20 subsets (A = 10, B = 11, C = 12) are the following. The common cardinality of sets in column n is A085089(n).
  1  12  13  124  145  1246  1467  12468  12689  1248A  1468B  12468C
             134       1346        13468  13689  1289A  148AB  1248AC
                       1456        14568  15689  1348A  1689B  12689C
                                   14678  16789  1389A  189AB  1289AC
                                                 1458A         13468C
                                                 1478A         1348AC
                                                 1589A         13689C
                                                 1789A         1389AC
                                                               14568C
                                                               1458AC
                                                               14678C
                                                               1468BC
                                                               1478AC
                                                               148ABC
                                                               15689C
                                                               1589AC
                                                               16789C
                                                               1689BC
                                                               1789AC
                                                               189ABC

Mathematica

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

Crossrefs

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

Sequence in context: A175126 A142884 A347028 * A177687 A219356 A142878

Adjacent sequences: A325362 A325363 A325364 * A325366 A325367 A325368

Keyword

nonn

Author

Gus Wiseman, Jul 06 2019

Status

approved