cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Aug 19 2018

Keywords

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[].

Examples

			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]]

Links

Crossrefs

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

Programs

  • 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[#],_[]]&]
  • Python
    See Neder link.

Extensions

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

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

Views

Author

Gus Wiseman, Aug 19 2018

Keywords

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.

Examples

			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))

Crossrefs

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

Programs

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

Views

Author

Gus Wiseman, Aug 19 2018

Keywords

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.

Examples

			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.

Crossrefs

Cf. A000081, A007853, A007916, A052409, A052410, A277576, A277996.
Cf. A317658, A316112, A317056, A317765, A317994, A318149, A318150, A318152.

Programs

  • 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]

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.

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

Views

Author

Gus Wiseman, Aug 19 2018

Keywords

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.

Crossrefs

A subsequence of A000079.
Cf. A000081, A007916, A029856, A052409, A052410, A277576, A277996, A280000.
Cf. A317658, A316112, A317056, A317765, A317994, A318149, A318150, A318152, A318153.
Showing 1-4 of 4 results.

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