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

Author

Gus Wiseman, Aug 19 2018

Keywords

• nonn

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

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

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