A317658 -id:A317658 - cp's OEIS frontend

A318149 e-numbers of free pure symmetric multifunctions with one atom.

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

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 empty subexpressions f[].

			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]]

Charlie Neder, Table of n, a(n) for n = 1..310
Charlie Neder, Python program for calculating this sequence

A subsequence of A001597.
Cf. A007916, A052409, A052410, A277576, A277996, A280000.
Cf. A317658, A316112, A317056, A317765, A317994, A318150, A318152, A318153.

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[#],_[]]&]

Python
```
See Neder link.
```

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

A318150 e-numbers of free pure functions with one atom.

Original entry on oeis.org

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

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

			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]

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

A subsequence of A001597.
Cf. A000108, A007916, A052409, A052410, A277576, A277996, A280000.
Cf. A317658, A316112, A317056, A317765, A317994, A318149, A318152, A318153.

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

More terms from Charlie Neder, Feb 23 2019

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.

Original entry on oeis.org

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

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 empty subexpressions f[] or subexpressions in heads f[x_1, ..., x_k][y_1, ..., y_k] where k,j >= 0.

			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))

A subsequence of A000079 and A318151.
Cf. A000081, A007916, A052409, A052410, A277576, A277996, A280000.
Cf. A317658, A316112, A317056, A317765, A317994, A318149, A318150, A318153.

baQ[n_]:=Or[n==1,MatchQ[FactorInteger[n],{{2,?(And@@Cases[FactorInteger[#],{p,k_}:>baQ[PrimePi[p]]]&)}}]];
Select[2^Range[0,50],baQ]

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

Original entry on oeis.org

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.

A317676 Triangle whose n-th row lists in order all e-numbers of free pure symmetric multifunctions (with empty expressions allowed) with one atom and n positions.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 16, 7, 10, 12, 13, 21, 25, 27, 32, 36, 64, 81, 128, 256, 11, 14, 17, 18, 28, 33, 35, 41, 45, 49, 75, 93, 100, 125, 144, 145, 169, 216, 243, 279, 441, 512, 625, 729, 1024, 1296, 2048, 2187, 4096, 6561, 8192, 16384, 65536, 524288, 8388608, 9007199254740992

Offset: 1

Gus Wiseman, Aug 03 2018

Given a positive integer n we construct a unique free pure symmetric multifunction e(n) 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)].

Every free pure symmetric multifunction (with empty expressions allowed) f with one atom and n positions has a unique e-number n such that e(n) = f, and vice versa, so this sequence is a permutation of the positive integers.

			Triangle begins:
  1
  2
  3   4
  5   6   8   9  16
  7  10  12  13  21  25  27  32  36  64  81 128 256
Corresponding triangle of free pure symmetric multifunctions (with empty expressions allowed) begins:
  o,
  o[],
  o[][], o[o],
  o[][][], o[o][], o[o[]], o[][o], o[o,o].

Mathematica Reference, Orderless

Row lengths are A277996.

Cf. A007916, A052409, A052410, A052893, A053492, A215366, A279944, A280000, A299759, A317658, A317659, A317677.

maxUsing[n_]:=If[n==1,{"o"},Join@@Cases[Table[PR[k,n-k-1],{k,n-1}],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{maxUsing[h],Union[Sort/@Tuples[maxUsing/@p]]}],{p,IntegerPartitions[g]}]]];
radQ[n_]:=And[n>1,GCD@@FactorInteger[n][[All,2]]==1];
Clear[rad];rad[n_]:=rad[n]=If[n==0,1,NestWhile[#+1&,rad[n-1]+1,Not[radQ[#]]&]];
ungo[x_?AtomQ]:=1;ungo[h_[g___]]:=rad[ungo[h]]^(Times@@Prime/@ungo/@{g});
Table[Sort[ungo/@maxUsing[n]],{n,5}]

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.

cp's OEIS Frontend

A318149 e-numbers of free pure symmetric multifunctions with one atom.

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A318150 e-numbers of free pure functions with one atom.

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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.

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Mathematica

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

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Mathematica

Formula

A317676 Triangle whose n-th row lists in order all e-numbers of free pure symmetric multifunctions (with empty expressions allowed) with one atom and n positions.

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Mathematica

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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