A316112 Number of leaves in the free pure symmetric multifunction (with empty expressions allowed) with e-number n.

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 1, 3, 2, 2, 2, 2, 1, 2, 3, 2, 2, 2, 2, 2, 1, 2, 3, 3, 2, 2, 2, 2, 2, 1, 2, 3, 3, 2, 2, 2, 2, 2, 2, 1, 2, 3, 3, 2, 2, 2, 2, 2, 2, 1, 3, 2, 3, 3, 2, 2, 2, 2, 2, 2, 1, 3, 2, 3, 3, 2, 2, 3, 2, 2, 2, 2, 1, 3

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 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

			e(21025) = o[o[o]][o] has 4 leaves so a(21025) = 4.
		

Crossrefs

Cf. A007916, A052409, A052410, A109129, A277576, A277996, A300626, A316112, A317056, A317658, A317765, A317994.

Programs

Mathematica

nn=1000;
radQ[n_]:=If[n==1,False,GCD@@FactorInteger[n][[All,2]]==1];
rad[n_]:=rad[n]=If[n==0,1,NestWhile[#+1&,rad[n-1]+1,Not[radQ[#]]&]];
Clear[radPi];Set@@@Array[radPi[rad[#]]==#&,nn];
a[n_]:=If[n==1,1,With[{g=GCD@@FactorInteger[n][[All,2]]},a[radPi[Power[n,1/g]]]+Sum[a[PrimePi[pr[[1]]]]*pr[[2]],{pr,If[g==1,{},FactorInteger[g]]}]]];
Table[a[n],{n,100}]

Formula

a(rad(x)^(prime(y_1) * ... * prime(y_k))) = a(x) + a(y_1) + ... + a(y_k) where rad = A007916.