Search: a316112 -id:a316112
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A317056
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Depth of the free pure symmetric multifunction (with empty expressions allowed) with e-number n.
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+10
9
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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
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OFFSET
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1,3
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COMMENTS
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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).
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LINKS
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EXAMPLE
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e(21025) = o[o[o]][o] has depth 3 so a(21025) = 3.
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MATHEMATICA
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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}]
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CROSSREFS
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Cf. A007916, A052409, A052410, A109082, A277576, A277996, A300626, A316112, A317056, A317658, A317765, A317994.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A317994
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Number of inequivalent leaf-colorings of the free pure symmetric multifunction with e-number n.
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+10
9
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1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 4, 2, 2, 2, 1, 4, 2, 2, 2, 2, 1, 2, 4, 2, 2, 2, 2, 2, 1, 2, 5, 4, 2, 2, 2, 2, 2, 1, 2, 5, 4, 2, 2, 2, 2, 2, 2, 1, 2, 5, 4, 2, 2, 2, 2, 2, 2, 1, 5, 2, 5, 4, 2, 2, 2, 2, 2, 2, 1, 5, 2, 5, 4, 2, 2, 4, 2, 2, 2, 2, 1, 5
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OFFSET
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1,4
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COMMENTS
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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).
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LINKS
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EXAMPLE
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Inequivalent representatives of the a(441) = 11 colorings of the expression e(441) = o[o,o][o] are the following.
1[1,1][1]
1[1,1][2]
1[1,2][1]
1[1,2][2]
1[1,2][3]
1[2,2][1]
1[2,2][2]
1[2,2][3]
1[2,3][1]
1[2,3][2]
1[2,3][4]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A317765
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Number of distinct subexpressions of the free pure symmetric multifunction (with empty expressions allowed) with e-number n.
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+10
8
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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
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OFFSET
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1,2
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COMMENTS
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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).
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LINKS
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EXAMPLE
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The a(12) = 4 subexpressions of o[o[]][] are {o, o[], o[o[]], o[o[]][]}.
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MATHEMATICA
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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}]
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CROSSREFS
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Cf. A007916, A052409, A052410, A277576, A277996, A300626, A316112, A317056, A317658, A317713, A317994.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A318149
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e-numbers of free pure symmetric multifunctions with one atom.
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+10
5
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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;
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OFFSET
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1,2
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COMMENTS
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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[].
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LINKS
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EXAMPLE
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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]]
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MATHEMATICA
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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[#], _[]]&]
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PROG
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(Python) See Neder link.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A318150
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e-numbers of free pure functions with one atom.
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+10
5
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1, 4, 36, 128, 2025, 21025, 279936, 4338889, 449482401, 78701569444, 373669453125, 18845583322500, 1347646586640625, 202054211912421649, 6193981883008128893161, 139629322539586311507076, 170147232533595290155627, 355156175404848064835984400
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graph;
refs;
listen;
history;
text;
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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a(1) = 1, and if a and b are in this sequence then so is rad(a)^prime(b). - Charlie Neder, Feb 23 2019
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EXAMPLE
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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]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A318152
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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.
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+10
4
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1, 4, 16, 128, 256, 16384, 65536, 268435456, 4294967296, 562949953421312, 9007199254740992, 72057594037927936, 18446744073709551616, 316912650057057350374175801344, 81129638414606681695789005144064, 5192296858534827628530496329220096
(list;
graph;
refs;
listen;
history;
text;
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OFFSET
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1,2
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COMMENTS
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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[] or subexpressions in heads f[x_1, ..., x_k][y_1, ..., y_k] where k,j >= 0.
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LINKS
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EXAMPLE
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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))
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MATHEMATICA
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baQ[n_]:=Or[n==1, MatchQ[FactorInteger[n], {{2, _?(And@@Cases[FactorInteger[#], {p_, k_}:>baQ[PrimePi[p]]]&)}}]];
Select[2^Range[0, 50], baQ]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A318153
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Number of antichain covers of the free pure symmetric multifunction (with empty expressions allowed) with e-number n.
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+10
4
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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;
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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MATHEMATICA
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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]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A318151
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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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+10
1
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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)
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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