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A316112 Number of leaves in the free pure symmetric multifunction (with empty expressions allowed) with e-number n. 9
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 (list; graph; refs; listen; history; text; internal format)
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: A163109 A286574 A329320 * A317994 A128428 A056171

Adjacent sequences:  A316109 A316110 A316111 * A316113 A316114 A316115

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 18 2018

STATUS

approved

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Last modified August 13 08:27 EDT 2020. Contains 336442 sequences. (Running on oeis4.)