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A277576 a(1)=1; thereafter a(n) = A007916(a(n-1)).

%I A277576 #21 Nov 22 2024 01:49:57
%S A277576 1,2,3,5,7,11,15,20,26,34,43,53,63,74,86,98,111,126,142,159,177,195,
%T A277576 214,235,258,281,305,330,356,383,411,439,468,498,530,562,595,629,663,
%U A277576 698,734,770,807,845,883,922,962,1003,1045,1087,1130,1174,1218,1263,1309,1356,1404,1453,1502,1552,1603,1654,1706,1759
%N A277576 a(1)=1; thereafter a(n) = A007916(a(n-1)).
%C A277576 Non-perfect-powers (A007916) are numbers such that the exponents in their prime factorizations have GCD equal to 1. For each n we can construct a plane tree by replacing all positive integers at any level with their corresponding planar factorization sequences (A277564), and repeating this replacement until no numbers are left. The result will be a unique "pure" sequence or plane tree. Under this correspondence a(n) is the path tree ((((((...)))))) = string of n consecutive open brackets followed by the same number of closed brackets.
%H A277576 Chai Wah Wu, <a href="/A277576/b277576.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..2480 from Gus Wiseman)
%e A277576 The first forty plane trees:
%e A277576 ()         11(((((())))))      ((()()()))         (((((((()())))))))
%e A277576 2(())         ((()(())))        ((((()(())))))     (()((())))
%e A277576 3((()))       (((())()))        (((((())()))))     ((((()))()))
%e A277576 (()())       ((((()()))))      ((((((()())))))) 34(((((((((())))))))))
%e A277576 5(((())))   15((((((()))))))    (((()))())         (((())(())))
%e A277576 ((()()))     (()()())        26((((((((())))))))) ((()())())
%e A277576 7((((()))))   (((()(()))))      ((())(()))         ((((()()()))))
%e A277576 (()(()))     ((((())())))      (((()()())))       ((((((()(())))))))
%e A277576 ((())())     (((((()())))))    (((((()(()))))))   (((((((())()))))))
%e A277576 (((()()))) 20(((((((())))))))  ((((((())())))))   ((((((((()()))))))))
%t A277576 radicalQ[1]:=False;radicalQ[n_]:=SameQ[GCD@@FactorInteger[n][[All,2]],1];
%t A277576 rad[0]:=1;rad[n_?Positive]:=rad[n]=NestWhile[#+1&,rad[n-1]+1,Not[radicalQ[#]]&];
%t A277576 nn=2000;Scan[rad,Range[nn]];NestWhileList[rad,1,#<nn&]
%o A277576 (Python)
%o A277576 from itertools import islice
%o A277576 from sympy import mobius, integer_nthroot
%o A277576 def A277576_gen(): # generator of terms
%o A277576     def iterfun(f,n=0):
%o A277576         m, k = n, f(n)
%o A277576         while m != k: m, k = k, f(k)
%o A277576         return m
%o A277576     def f(x): return int(1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
%o A277576     a = 1
%o A277576     while True:
%o A277576         yield a
%o A277576         a = iterfun(lambda x:f(x)+a,a)
%o A277576 A277576_list = list(islice(A277576_gen(),40)) # _Chai Wah Wu_, Nov 21 2024
%Y A277576 Cf. A007916, A277564, A276625, A004111 (rooted trees), A007097 (rooted paths).
%K A277576 nonn
%O A277576 1,2
%A A277576 _Gus Wiseman_, Oct 20 2016
%E A277576 Edited by _N. J. A. Sloane_, Nov 09 2016