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e-numbers of free pure symmetric multifunctions with one atom.
5

%I #14 Jan 01 2021 18:13:24

%S 1,4,16,36,128,256,441,1296,2025,16384,21025,65536,77841,194481,

%T 220900,279936,1679616,1803649,4100625,4338889,268435456,273571600,

%U 442050625,449482401,1801088541,4294967296,4334247225,6059221281

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

%C 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[].

%H Charlie Neder, <a href="/A318149/b318149.txt">Table of n, a(n) for n = 1..310</a>

%H Charlie Neder, <a href="/A318149/a318149.py.txt">Python program for calculating this sequence</a>

%e The sequence of free pure symmetric multifunctions with one atom "o", together with their e-numbers begins:

%e 1: o

%e 4: o[o]

%e 16: o[o,o]

%e 36: o[o][o]

%e 128: o[o[o]]

%e 256: o[o,o,o]

%e 441: o[o,o][o]

%e 1296: o[o][o,o]

%e 2025: o[o][o][o]

%e 16384: o[o,o[o]]

%e 21025: o[o[o]][o]

%e 65536: o[o,o,o,o]

%e 77841: o[o,o,o][o]

%e 194481: o[o,o][o,o]

%e 220900: o[o,o][o][o]

%e 279936: o[o][o[o]]

%t nn=1000;

%t radQ[n_]:=If[n==1,False,GCD@@FactorInteger[n][[All,2]]==1];

%t rad[n_]:=rad[n]=If[n==0,1,NestWhile[#+1&,rad[n-1]+1,Not[radQ[#]]&]];

%t Clear[radPi];Set@@@Array[radPi[rad[#]]==#&,nn];

%t 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]]]]]];

%t Select[Range[nn],FreeQ[exp[#],_[]]&]

%o (Python) See Neder link.

%Y A subsequence of A001597.

%Y Cf. A007916, A052409, A052410, A277576, A277996, A280000.

%Y Cf. A317658, A316112, A317056, A317765, A317994, A318150, A318152, A318153.

%K nonn

%O 1,2

%A _Gus Wiseman_, Aug 19 2018

%E a(16)-a(27) from _Charlie Neder_, Sep 01 2018