A317994 Number of inequivalent leaf-colorings of the free pure symmetric multifunction with e-number n.

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 4, 2, 2, 2, 1, 4, 2, 2, 2, 2, 1, 2, 4, 2, 2, 2, 2, 2, 1, 2, 5, 4, 2, 2, 2, 2, 2, 1, 2, 5, 4, 2, 2, 2, 2, 2, 2, 1, 2, 5, 4, 2, 2, 2, 2, 2, 2, 1, 5, 2, 5, 4, 2, 2, 2, 2, 2, 2, 1, 5, 2, 5, 4, 2, 2, 4, 2, 2, 2, 2, 1, 5

Offset: 1

Author

Gus Wiseman, Aug 18 2018

Keywords

• nonn

Comments

If n = 1 let e(n) be the leaf symbol "o". Given a positive integer n > 1 we construct a unique free pure symmetric multifunction (with empty expressions allowed) e(n) with one atom by expressing n as a power of a number that is not a perfect power to a product of prime numbers: n = rad(x)^(prime(y_1) * ... * prime(y_k)) where rad = A007916. Then e(n) = e(x)[e(y_1), ..., e(y_k)]. For example, e(21025) = o[o[o]][o] because 21025 = rad(rad(1)^prime(rad(1)^prime(1)))^prime(1).

Examples

			Inequivalent representatives of the a(441) = 11 colorings of the expression e(441) = o[o,o][o] are the following.
  1[1,1][1]
  1[1,1][2]
  1[1,2][1]
  1[1,2][2]
  1[1,2][3]
  1[2,2][1]
  1[2,2][2]
  1[2,2][3]
  1[2,3][1]
  1[2,3][2]
  1[2,3][4]
		

Crossrefs

Cf. A007916, A052409, A052410, A277576, A277996, A300626, A316112, A317056, A317658, A317765.