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

%I A318149 #14 Jan 01 2021 18:13:24
%S A318149 1,4,16,36,128,256,441,1296,2025,16384,21025,65536,77841,194481,
%T A318149 220900,279936,1679616,1803649,4100625,4338889,268435456,273571600,
%U A318149 442050625,449482401,1801088541,4294967296,4334247225,6059221281
%N A318149 e-numbers of free pure symmetric multifunctions with one atom.
%C A318149 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 A318149 Charlie Neder, <a href="/A318149/b318149.txt">Table of n, a(n) for n = 1..310</a>
%H A318149 Charlie Neder, <a href="/A318149/a318149.py.txt">Python program for calculating this sequence</a>
%e A318149 The sequence of free pure symmetric multifunctions with one atom "o", together with their e-numbers begins:
%e A318149        1: o
%e A318149        4: o[o]
%e A318149       16: o[o,o]
%e A318149       36: o[o][o]
%e A318149      128: o[o[o]]
%e A318149      256: o[o,o,o]
%e A318149      441: o[o,o][o]
%e A318149     1296: o[o][o,o]
%e A318149     2025: o[o][o][o]
%e A318149    16384: o[o,o[o]]
%e A318149    21025: o[o[o]][o]
%e A318149    65536: o[o,o,o,o]
%e A318149    77841: o[o,o,o][o]
%e A318149   194481: o[o,o][o,o]
%e A318149   220900: o[o,o][o][o]
%e A318149   279936: o[o][o[o]]
%t A318149 nn=1000;
%t A318149 radQ[n_]:=If[n==1,False,GCD@@FactorInteger[n][[All,2]]==1];
%t A318149 rad[n_]:=rad[n]=If[n==0,1,NestWhile[#+1&,rad[n-1]+1,Not[radQ[#]]&]];
%t A318149 Clear[radPi];Set@@@Array[radPi[rad[#]]==#&,nn];
%t A318149 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 A318149 Select[Range[nn],FreeQ[exp[#],_[]]&]
%o A318149 (Python) See Neder link.
%Y A318149 A subsequence of A001597.
%Y A318149 Cf. A007916, A052409, A052410, A277576, A277996, A280000.
%Y A318149 Cf. A317658, A316112, A317056, A317765, A317994, A318150, A318152, A318153.
%K A318149 nonn
%O A318149 1,2
%A A318149 _Gus Wiseman_, Aug 19 2018
%E A318149 a(16)-a(27) from _Charlie Neder_, Sep 01 2018