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A277615 a(1)=1; thereafter, if n = c(x_1)^...^c(x_k) (where c(k) = A007916(k) and with parentheses nested from the right, as in the definition of A277564), a(n) = 1 + a(x_1) + ... + a(x_k).

%I A277615 #22 Nov 05 2018 21:06:25
%S A277615 1,2,3,3,4,4,5,4,4,5,6,5,5,6,7,4,6,6,7,8,5,7,7,8,5,9,5,6,8,8,9,5,6,10,
%T A277615 6,5,7,9,9,10,6,7,11,7,6,8,10,10,6,11,7,8,12,8,7,9,11,11,7,12,8,9,13,
%U A277615 5,9,8,10,12,12,8,13,9,10,14,6,10,9,11,13,13,5,9,14,10,11,15,7,11,10,12,14,14,6,10,15,11,12
%N A277615 a(1)=1; thereafter, if n = c(x_1)^...^c(x_k) (where c(k) = A007916(k) and with parentheses nested from the right, as in the definition of A277564), a(n) = 1 + a(x_1) + ... + a(x_k).
%C A277615 A007916 lists the numbers whose prime multiplicities are relatively prime. For each n we can construct a plane tree by repeatedly factoring all positive integers at any level into their corresponding power towers of non-perfect-powers (see A277564). a(n) is the number of nodes in this plane tree.
%H A277615 Gus Wiseman, <a href="/A277615/b277615.txt">Table of n, a(n) for n = 1..10000</a>
%F A277615 First appearance of n is a(A277576(n)). Last appearance of n is a(2^^{n-1}) where ^^ denotes iterated exponentiation (or tetration).
%F A277615 Number of appearances of n is the Catalan number |{k:a(k)=n}| = C_{n-1}.
%e A277615 a(1)=1, a(2)=1+a(1)=2, a(3)=1+a(2)=3, a(4)=1+a(1)+a(1)=3 because 4=c(1)^c(1), a(8)=1+a(1)+a(2)=4 because 8=c(1)^c(2), a(9)=1+a(2)+a(1)=4 because 9=c(2)^c(1), a(10)=1+a(6)=5 because 10=c(6).
%t A277615 nn=10000;
%t A277615 radicalQ[1]:=False;radicalQ[n_]:=SameQ[GCD@@FactorInteger[n][[All,2]],1];
%t A277615 hyperfactor[1]:={};hyperfactor[n_?radicalQ]:={n};
%t A277615 hyperfactor[n_]:=With[{g=GCD@@FactorInteger[n][[All,2]]},Prepend[hyperfactor[g],Product[Apply[Power[#1,#2/g]&,r],{r,FactorInteger[n]}]]];
%t A277615 rad[0]:=1;rad[n_?Positive]:=rad[n]=NestWhile[#+1&,rad[n-1]+1,Not[radicalQ[#]]&];Set@@@Array[radPi[rad[#]]==#&,nn];
%t A277615 rnk[n_]:=rnk[n]=1+Total[rnk/@radPi/@hyperfactor[n]];
%t A277615 Array[rnk,nn]
%Y A277615 Cf. A000108, A007916, A014221, A277564, A277576.
%K A277615 nonn,look
%O A277615 1,2
%A A277615 _Gus Wiseman_, Oct 23 2016
%E A277615 Edited by _N. J. A. Sloane_, Nov 09 2016