A277576 -id:A277576 - cp's OEIS frontend

A318153 Number of antichain covers of the free pure symmetric multifunction (with empty expressions allowed) with e-number n.

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

Offset: 1

Gus Wiseman, Aug 19 2018

nonn

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.

			441 is the e-number of o[o,o][o] which has antichain covers {o[o,o][o]}, {o[o,o], o}, {o, o, o, o}}, corresponding to the leaf-colorings 1[1,1][1], 1[1,1][2], 1[2,3][4], so a(441) = 3.

Cf. A000081, A007853, A007916, A052409, A052410, A277576, A277996.

Cf. A317658, A316112, A317056, A317765, A317994, A318149, A318150, A318152.

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[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]]},1+a[radPi[n^(1/g)]]*Product[a[PrimePi[pr[[1]]]]^pr[[2]],{pr,If[g==1,{},FactorInteger[g]]}]]];
Array[a,100]

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.

A318151 e-numbers of unlabeled rooted trees with empty leaves o[] allowed. A number n is in the sequence iff n = 2^(prime(y_1) * ... * prime(y_k)) for some k >= 0 and y_1, ..., y_k already in the sequence.

Original entry on oeis.org

1, 2, 4, 8, 16, 64, 128, 256, 512, 4096, 16384, 65536, 262144, 524288, 2097152, 16777216, 134217728, 268435456, 4294967296, 68719476736, 274877906944, 4398046511104, 281474976710656, 562949953421312, 9007199254740992, 18014398509481984, 72057594037927936

Offset: 1

Gus Wiseman, Aug 19 2018

nonn

If n = 1 let e(n) be the leaf symbol "o". Given a positive integer n > 1 we construct a unique orderless expression 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 sequence consists of all numbers n such that e(n) contains no subexpressions in heads f[x_1, ..., x_k][y_1, ..., y_k] where k,j >= 0.

A subsequence of A000079.
Cf. A000081, A007916, A029856, A052409, A052410, A277576, A277996, A280000.
Cf. A317658, A316112, A317056, A317765, A317994, A318149, A318150, A318152, A318153.

A292127 a(1) = 1, a(r(n)^k) = 1 + k * a(n) where r(n) is the n-th number that is not a perfect power A007916(n).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 09 2017

nonn

Any positive integer greater than 1 can be written uniquely as a perfect power r(n)^k. We define a planted achiral (or generalized Bethe) tree b(n) for any positive integer greater than 1 by writing n as a perfect power r(d)^k and forming a tree with k branches all equal to b(d). Then a(n) is the number of nodes in b(n).

			The first nineteen planted achiral trees are:
o,
(o),
((o)), (oo),
(((o))), ((oo)),
((((o)))), (ooo), ((o)(o)), (((oo))),
(((((o))))), ((ooo)), (((o)(o))), ((((oo)))),
((((((o)))))), (oooo), (((ooo))), ((((o)(o)))), (((((oo))))).

Cf. A003238, A007916, A052409, A052410, A061775, A214577, A277576, A277615, A278028, A279614, A279944, A289023.

nn=100;
rads=Select[Range[2,nn],GCD@@FactorInteger[#][[All,2]]===1&];
a[1]:=1;a[n_]:=With[{k=GCD@@FactorInteger[n][[All,2]]},1+k*a[Position[rads,n^(1/k)][[1,1]]]];
Array[a,nn]

cp's OEIS Frontend

A318153 Number of antichain covers of the free pure symmetric multifunction (with empty expressions allowed) with e-number n.

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A318151 e-numbers of unlabeled rooted trees with empty leaves o[] allowed. A number n is in the sequence iff n = 2^(prime(y_1) * ... * prime(y_k)) for some k >= 0 and y_1, ..., y_k already in the sequence.

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A292127 a(1) = 1, a(r(n)^k) = 1 + k * a(n) where r(n) is the n-th number that is not a perfect power A007916(n).

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Mathematica