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Search: a318153 -id:a318153
Displaying 1-4 of 4 results found.
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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
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.
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
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
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.
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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