A317994 -id:A317994

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A316112

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

+10
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.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 18 2018

STATUS

approved

A317056

Depth of the free pure symmetric multifunction (with empty expressions allowed) with e-number n.

+10
9

0, 1, 2, 1, 3, 2, 4, 2, 2, 3, 5, 3, 3, 4, 6, 1, 4, 4, 5, 7, 2, 5, 5, 6, 3, 8, 2, 3, 6, 6, 7, 3, 4, 9, 3, 2, 4, 7, 7, 8, 4, 5, 10, 4, 3, 5, 8, 8, 4, 9, 5, 6, 11, 5, 4, 6, 9, 9, 5, 10, 6, 7, 12, 2, 6, 5, 7, 10, 10, 6, 11, 7, 8, 13, 3, 7, 6, 8, 11, 11, 2, 7, 12

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

LINKS

Table of n, a(n) for n=1..83.

EXAMPLE

e(21025) = o[o[o]][o] has depth 3 so a(21025) = 3.

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];

exp[n_]:=If[n===1, "o", With[{g=GCD@@FactorInteger[n][[All, 2]]}, Apply[exp[radPi[Power[n, 1/g]]], exp/@Flatten[Cases[FactorInteger[g], {p_?PrimeQ, k_}:>ConstantArray[PrimePi[p], k]]]]]];

Table[Max@@Length/@Position[exp[n], _], {n, 200}]

CROSSREFS

Cf. A007916, A052409, A052410, A109082, A277576, A277996, A300626, A316112, A317056, A317658, A317765, A317994.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 18 2018

STATUS

approved

A317765

Number of distinct subexpressions of the free pure symmetric multifunction (with empty expressions allowed) with e-number n.

+10
8

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

(list; graph; refs; listen; history; text; internal format)

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) 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..82.

EXAMPLE

The a(12) = 4 subexpressions of o[o[]][] are {o, o[], o[o[]], o[o[]][]}.

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];

exp[n_]:=If[n===1, "o", With[{g=GCD@@FactorInteger[n][[All, 2]]}, Apply[exp[radPi[Power[n, 1/g]]], exp/@Flatten[Cases[FactorInteger[g], {p_?PrimeQ, k_}:>ConstantArray[PrimePi[p], k]]]]]];

Table[Length[Union[Cases[exp[n], _, {0, Infinity}, Heads->True]]], {n, 100}]

CROSSREFS

Cf. A007916, A052409, A052410, A277576, A277996, A300626, A316112, A317056, A317658, A317713, A317994.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 18 2018

STATUS

approved

A304486

Number of inequivalent leaf-colorings of the unlabeled rooted tree with Matula-Goebel number n.

+10
7

1, 1, 1, 2, 1, 2, 2, 3, 2, 2, 1, 4, 2, 4, 2, 5, 2, 4, 3, 4, 4, 2, 2, 7, 2, 5, 3, 9, 2, 5, 1, 7, 2, 4, 4, 9, 4, 7, 5, 7, 2, 11, 4, 4, 4, 4, 2, 12, 7, 4, 4, 11, 5, 7, 2, 16, 7, 5, 2, 11, 4, 2, 9, 11, 5, 5, 3, 9, 4, 11

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

Table of n, a(n) for n=1..70.

EXAMPLE

Inequivalent representatives of the a(52) = 11 colorings of the tree (oo(o(o))) are the following.

(11(1(1)))

(11(1(2)))

(11(2(1)))

(11(2(2)))

(11(2(3)))

(12(1(1)))

(12(1(2)))

(12(1(3)))

(12(3(1)))

(12(3(3)))

(12(3(4)))

CROSSREFS

Cf. A000081, A001678, A004111, A007097, A049076, A061773, A061775, A109082, A109129, A300626, A303024, A303431, A317713, A317994.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Aug 17 2018

STATUS

approved

A318149

e-numbers of free pure symmetric multifunctions with one atom.

+10
5

1, 4, 16, 36, 128, 256, 441, 1296, 2025, 16384, 21025, 65536, 77841, 194481, 220900, 279936, 1679616, 1803649, 4100625, 4338889, 268435456, 273571600, 442050625, 449482401, 1801088541, 4294967296, 4334247225, 6059221281

(list; graph; refs; listen; history; text; internal format)

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 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 empty subexpressions f[].

LINKS

Charlie Neder, Table of n, a(n) for n = 1..310

Charlie Neder, Python program for calculating this sequence

EXAMPLE

The sequence of free pure symmetric multifunctions with one atom "o", together with their e-numbers begins:

1: o

4: o[o]

16: o[o,o]

36: o[o][o]

128: o[o[o]]

256: o[o,o,o]

441: o[o,o][o]

1296: o[o][o,o]

2025: o[o][o][o]

16384: o[o,o[o]]

21025: o[o[o]][o]

65536: o[o,o,o,o]

77841: o[o,o,o][o]

194481: o[o,o][o,o]

220900: o[o,o][o][o]

279936: o[o][o[o]]

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];

exp[n_]:=If[n==1, "o", With[{g=GCD@@FactorInteger[n][[All, 2]]}, Apply[exp[radPi[Power[n, 1/g]]], exp/@Flatten[Cases[FactorInteger[g], {p_?PrimeQ, k_}:>ConstantArray[PrimePi[p], k]]]]]];

Select[Range[nn], FreeQ[exp[#], _[]]&]

PROG

(Python) See Neder link.

CROSSREFS

A subsequence of A001597.

Cf. A007916, A052409, A052410, A277576, A277996, A280000.

Cf. A317658, A316112, A317056, A317765, A317994, A318150, A318152, A318153.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 19 2018

EXTENSIONS

a(16)-a(27) from Charlie Neder, Sep 01 2018

STATUS

approved

A318150

e-numbers of free pure functions with one atom.

+10
5

1, 4, 36, 128, 2025, 21025, 279936, 4338889, 449482401, 78701569444, 373669453125, 18845583322500, 1347646586640625, 202054211912421649, 6193981883008128893161, 139629322539586311507076, 170147232533595290155627, 355156175404848064835984400

(list; graph; refs; listen; history; text; internal format)

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 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). This sequence consists of all numbers n such that e(n) contains no non-unitary subexpressions f[x_1, ..., x_k] where k != 1.

LINKS

Charlie Neder, Table of n, a(n) for n = 1..44

FORMULA

a(1) = 1, and if a and b are in this sequence then so is rad(a)^prime(b). - Charlie Neder, Feb 23 2019

EXAMPLE

The sequence of all free pure functions with one atom together with their e-numbers begins:

1: o

4: o[o]

36: o[o][o]

128: o[o[o]]

2025: o[o][o][o]

21025: o[o[o]][o]

279936: o[o][o[o]]

4338889: o[o][o][o][o]

CROSSREFS

A subsequence of A001597.

Cf. A000108, A007916, A052409, A052410, A277576, A277996, A280000.

Cf. A317658, A316112, A317056, A317765, A317994, A318149, A318152, A318153.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 19 2018

EXTENSIONS

More terms from Charlie Neder, Feb 23 2019

STATUS

approved

A318152

e-numbers of unlabeled rooted trees. 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.

+10
4

1, 4, 16, 128, 256, 16384, 65536, 268435456, 4294967296, 562949953421312, 9007199254740992, 72057594037927936, 18446744073709551616, 316912650057057350374175801344, 81129638414606681695789005144064, 5192296858534827628530496329220096

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

LINKS

Table of n, a(n) for n=1..16.

EXAMPLE

The sequence contains 16384 = 2^14 = 2^(prime(1) * prime(4)) because 1 and 4 both already belong to the sequence.

The sequence of unlabeled rooted trees with e-numbers in the sequence begins:

1: o

4: (o)

16: (oo)

128: ((o))

256: (ooo)

16384: (o(o))

65536: (oooo)

. (oo(o))

. (ooooo)

. ((o)(o))

((oo))

(ooo(o))

(oooooo)

(o(o)(o))

(o(oo))

(oooo(o))

(ooooooo)

(oo(o)(o))

MATHEMATICA

baQ[n_]:=Or[n==1, MatchQ[FactorInteger[n], {{2, _?(And@@Cases[FactorInteger[#], {p_, k_}:>baQ[PrimePi[p]]]&)}}]];

Select[2^Range[0, 50], baQ]

CROSSREFS

A subsequence of A000079 and A318151.

Cf. A000081, A007916, A052409, A052410, A277576, A277996, A280000.

Cf. A317658, A316112, A317056, A317765, A317994, A318149, A318150, A318153.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 19 2018

STATUS

approved

A318153

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

+10
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

(list; graph; refs; listen; history; text; internal format)

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 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.

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[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]

CROSSREFS

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

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

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 19 2018

STATUS

approved

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.

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

LINKS

Table of n, a(n) for n=1..27.

CROSSREFS

A subsequence of A000079.

Cf. A000081, A007916, A029856, A052409, A052410, A277576, A277996, A280000.

Cf. A317658, A316112, A317056, A317765, A317994, A318149, A318150, A318152, A318153.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 19 2018

STATUS

approved

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