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

1,4

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).

Links

Table of n, a(n) for n=1..87.

Formula

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

Example

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

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}]

Crossrefs

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

Sequence in context: A359289 A286574 A329320 * A317994 A128428 A056171

Adjacent sequences: A316109 A316110 A316111 * A316113 A316114 A316115

Keyword

nonn

Author

Gus Wiseman, Aug 18 2018

Status

approved