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A304481 Turn the power-tower for n upside-down.

%I A304481 #12 May 14 2018 03:09:55
%S A304481 1,2,3,4,5,6,7,9,8,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,32,26,
%T A304481 27,28,29,30,31,25,33,34,35,64,37,38,39,40,41,42,43,44,45,46,47,48,
%U A304481 128,50,51,52,53,54,55,56,57,58,59,60,61,62,63,36,65,66,67
%N A304481 Turn the power-tower for n upside-down.
%C A304481 This is an involution of the positive integers.
%C A304481 The power-tower for n is defined as follows. Let {c(i)} = A007916 denote the sequence of numbers > 1 which are not perfect powers. Every positive integer n has a unique representation as a tower n = c(x_1)^c(x_2)^c(x_3)^...^c(x_k), where the exponents are nested from the right. Then a(n) = c(x_k)^...^c(x_3)^c(x_2)^c(x_1).
%H A304481 Robert Israel, <a href="/A304481/b304481.txt">Table of n, a(n) for n = 1..10000</a>
%H A304481 N. J. A. Sloane, <a href="/A278028/a278028.txt">Maple programs for A007916, A278028, A278029, A052409, A089723, A277564</a>
%e A304481 The power tower of 81 is 3^2^2, which turned upside-down is 2^2^3 = 256, so a(81) = 256.
%p A304481 f:= proc(n,r) local F,a,y;
%p A304481      if n = 1 then return 1 fi;
%p A304481      F:= ifactors(n)[2];
%p A304481      y:= igcd(seq(t[2],t=F));
%p A304481      if y = 1 then return n^r fi;
%p A304481      a:= mul(t[1]^(t[2]/y),t=F);
%p A304481      procname(y,a^r)
%p A304481 end proc:
%p A304481 seq(f(n,1),n=1..100); # _Robert Israel_, May 13 2018
%t A304481 tow[n_]:=If[n==1,{},With[{g=GCD@@FactorInteger[n][[All,2]]},If[g===1,{n},Prepend[tow[g],n^(1/g)]]]];
%t A304481 Table[Power@@Reverse[tow[n]],{n,100}]
%Y A304481 Cf. A052409, A052410, A007916, A089723, A164337, A277562, A277564, A278028, A288636, A289023, A294336, A294337, A304491, A304492, A304495.
%K A304481 nonn,look
%O A304481 1,2
%A A304481 _Gus Wiseman_, May 13 2018