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A277576
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a(1)=1; thereafter a(n) = A007916(a(n-1)).
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14
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1, 2, 3, 5, 7, 11, 15, 20, 26, 34, 43, 53, 63, 74, 86, 98, 111, 126, 142, 159, 177, 195, 214, 235, 258, 281, 305, 330, 356, 383, 411, 439, 468, 498, 530, 562, 595, 629, 663, 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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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The first forty plane trees:
() 11(((((()))))) ((()()())) (((((((()())))))))
2(()) ((()(()))) ((((()(()))))) (()((())))
3((())) (((())())) (((((())())))) ((((()))()))
(()()) ((((()())))) ((((((()())))))) 34(((((((((())))))))))
5(((()))) 15((((((())))))) (((()))()) (((())(())))
((()())) (()()()) 26((((((((())))))))) ((()())())
7((((())))) (((()(())))) ((())(())) ((((()()()))))
(()(())) ((((())()))) (((()()()))) ((((((()(())))))))
((())()) (((((()()))))) (((((()(())))))) (((((((())()))))))
(((()()))) 20(((((((()))))))) ((((((())()))))) ((((((((()()))))))))
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MATHEMATICA
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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&]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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