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A277576 a(1)=1; thereafter a(n) = A007916(a(n-1)). 5
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 (list; graph; refs; listen; history; text; internal format)
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

Gus Wiseman, Table of n, a(n) for n = 1..2480

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&]

CROSSREFS

Cf. A007916, A277564, A276625, A004111 (rooted trees), A007097 (rooted paths).

Sequence in context: A240556 A090693 A260794 * A260164 A280938 A049756

Adjacent sequences:  A277573 A277574 A277575 * A277577 A277578 A277579

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 20 2016

EXTENSIONS

Edited by N. J. A. Sloane, Nov 09 2016

STATUS

approved

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Last modified January 22 08:47 EST 2018. Contains 298042 sequences.