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A325263
Number of subsets of {1..n} containing n such that no two elements have the same sorted prime signature.
4
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
1,2
COMMENTS
The sorted prime signature (A118914) of a positive integer is the multiset of exponents in its standard factorization into prime numbers.
FORMULA
a(n) = A326438(n)/(1 + A064839(n)).
EXAMPLE
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}
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}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 06 2019
STATUS
approved