

A318153


Number of antichain covers of the free pure symmetric multifunction (with empty expressions allowed) with enumber n.


4



1, 2, 3, 2, 4, 3, 5, 3, 3, 4, 6, 4, 4, 5, 7, 2, 5, 5, 6, 8, 3, 6, 6, 7, 4, 9, 5, 4, 7, 7, 8, 4, 5, 10, 6, 3, 5, 8, 8, 9, 5, 6, 11, 7, 4, 6, 9, 9, 5, 10, 6, 7, 12, 8, 5, 7, 10, 10, 6, 11, 7, 8, 13, 3, 9, 6, 8, 11, 11, 7, 12, 8, 9, 14, 4, 10, 7, 9, 12, 12, 3, 8
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OFFSET

1,2


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) (as can be represented in functional programming languages such as Mathematica) 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). The a(n) is the number of ways to partition e(n) into disjoint subexpressions such that all leaves are covered by exactly one of them.


LINKS

Table of n, a(n) for n=1..82.


FORMULA

If n = rad(x)^(Product_i prime(y_i)^z_i) where rad = A007916 then a(n) = 1 + a(x) * Product_i a(y_i)^z_i.


EXAMPLE

441 is the enumber of o[o,o][o] which has antichain covers {o[o,o][o]}, {o[o,o], o}, {o, o, o, o}}, corresponding to the leafcolorings 1[1,1][1], 1[1,1][2], 1[2,3][4], so a(441) = 3.


MATHEMATICA

nn=20000;
radQ[n_]:=If[n==1, False, GCD@@FactorInteger[n][[All, 2]]==1];
rad[n_]:=rad[n]=If[n==0, 1, NestWhile[#+1&, rad[n1]+1, Not[radQ[#]]&]];
Clear[radPi]; Set@@@Array[radPi[rad[#]]==#&, nn];
a[n_]:=If[n==1, 1, With[{g=GCD@@FactorInteger[n][[All, 2]]}, 1+a[radPi[n^(1/g)]]*Product[a[PrimePi[pr[[1]]]]^pr[[2]], {pr, If[g==1, {}, FactorInteger[g]]}]]];
Array[a, 100]


CROSSREFS

Cf. A000081, A007853, A007916, A052409, A052410, A277576, A277996.
Cf. A317658, A316112, A317056, A317765, A317994, A318149, A318150, A318152.
Sequence in context: A286531 A331280 A317765 * A157893 A331253 A264116
Adjacent sequences: A318150 A318151 A318152 * A318154 A318155 A318156


KEYWORD

nonn


AUTHOR

Gus Wiseman, Aug 19 2018


STATUS

approved



