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%I #39 Jun 16 2017 22:39:20
%S 1,2,1,3,2,4,1,1,5,3,6,4,7,5,8,1,2,9,2,1,10,6,11,7,12,8,13,9,14,10,15,
%T 11,16,1,1,1,17,12,18,13,19,14,20,15,21,16,22,17,23,18,24,19,25,3,1,
%U 26,20,27,2,2,28,21,29,22,30,23,31,24,32,1,3,33,25,34,26,35,27,36,4,1,37,28,38,29,39,30,40,31
%N Let {c(i)} = A007916 denote the sequence of numbers > 1 which are not perfect powers. Every positive integer n has a unique representation as a tower n = c(x_1)^c(x_2)^c(x_3)^...^c(x_k), where the exponents are nested from the right. The sequence is an irregular triangle read by rows, where the n-th row lists n followed by x_1, ..., x_k.
%C The row lengths are A288636(n) + 1. - _Gus Wiseman_, Jun 12 2017
%C See A278028 for a version in which row n simply lists x_1, x_2, ..., x_k (omitting the initial n).
%H Gus Wiseman, <a href="/A277564/b277564.txt">Table of n, a(n) for n = 1..20131</a>
%H N. J. A. Sloane, <a href="/A278028/a278028.txt">Maple programs for A007916, A278028, A278029, A052409, A089723, A277564</a>
%e 1 is represented by the empty sequence (), by convention.
%e Successive rows of the triangle are as follows (c(k) denotes the k-th non-prime-power, A007916(k)):
%e 2, 1,
%e 3, 2,
%e 4, 1, 1,
%e 5, 3,
%e 6, 4, because 6 = c(4)
%e 7, 5,
%e 8, 1, 2, because 8 = 2^3 = c(1)^c(2)
%e 9, 2, 1,
%e 10, 6,
%e 11, 7,
%e ...
%e 16, 1, 1, 1, because 16 = 2^4 = c(1)^4 = c(1)^(c(1)^2) = c[1]^(c[1]^c[1])
%e 17, 12,
%e ...
%e This sequence represents a bijection N -> Q where Q is the set of all finite sequences of positive integers: 1->(), 2->(1), 3->(2), 4->(1 1), 5->(3), 6->(4), 7->(5), 8->(1 2), 9->(2 1), ...
%p See link.
%t nn=10000;radicalQ[1]:=False;radicalQ[n_]:=SameQ[GCD@@FactorInteger[n][[All,2]],1];
%t hyperfactor[1]:={};hyperfactor[n_?radicalQ]:={n};hyperfactor[n_]:=With[{g=GCD@@FactorInteger[n][[All,2]]},Prepend[hyperfactor[g],Product[Apply[Power[#1,#2/g]&,r],{r,FactorInteger[n]}]]];
%t rad[0]:=1;rad[n_?Positive]:=rad[n]=NestWhile[#+1&,rad[n-1]+1,Not[radicalQ[#]]&];Set@@@Array[radPi[rad[#]]==#&,nn];
%t Flatten[Join[{#},radPi/@hyperfactor[#]]&/@Range[nn]]
%Y Cf. A007916, A277562, A164336, A000961, A164337, A278028, A089723.
%K nonn,tabf
%O 1,2
%A _Gus Wiseman_, Oct 20 2016
%E Edited by _N. J. A. Sloane_, Nov 09 2016
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