OFFSET
1,2
COMMENTS
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.
LINKS
Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..2480 from Gus Wiseman)
EXAMPLE
The first forty plane trees:
() 11(((((()))))) ((()()())) (((((((()())))))))
2(()) ((()(()))) ((((()(()))))) (()((())))
3((())) (((())())) (((((())())))) ((((()))()))
(()()) ((((()())))) ((((((()())))))) 34(((((((((())))))))))
5(((()))) 15((((((())))))) (((()))()) (((())(())))
((()())) (()()()) 26((((((((())))))))) ((()())())
7((((())))) (((()(())))) ((())(())) ((((()()()))))
(()(())) ((((())()))) (((()()()))) ((((((()(())))))))
((())()) (((((()()))))) (((((()(())))))) (((((((())()))))))
(((()()))) 20(((((((()))))))) ((((((())()))))) ((((((((()()))))))))
MATHEMATICA
radicalQ[1]:=False; radicalQ[n_]:=SameQ[GCD@@FactorInteger[n][[All, 2]], 1];
rad[0]:=1; rad[n_?Positive]:=rad[n]=NestWhile[#+1&, rad[n-1]+1, Not[radicalQ[#]]&];
nn=2000; Scan[rad, Range[nn]]; NestWhileList[rad, 1, #<nn&]
PROG
(Python)
from itertools import islice
from sympy import mobius, integer_nthroot
def A277576_gen(): # generator of terms
def iterfun(f, n=0):
m, k = n, f(n)
while m != k: m, k = k, f(k)
return m
def f(x): return int(1-sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(2, x.bit_length())))
a = 1
while True:
yield a
a = iterfun(lambda x:f(x)+a, a)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 20 2016
EXTENSIONS
Edited by N. J. A. Sloane, Nov 09 2016
STATUS
approved