%I #27 May 01 2024 01:45:54
%S 16,81,256,512,625,1296,2401,6561,10000,14641,19683,20736,28561,38416,
%T 50625,65536,83521,104976,130321,160000,194481,234256,279841,331776,
%U 390625,456976,614656,707281,810000,923521,1185921,1336336,1500625,1679616,1874161,1953125,2085136,2313441,2560000,2825761,3111696,3418801
%N Numbers of the form c(x_1)^c(x_2)^...^c(x_k) where each c(i) = A007916(i) is a non-perfect-power, k >= 2, and the exponents are nested from the right.
%C Non-perfect-powers, or NPPs (A007916), are numbers whose prime multiplicities are relatively prime. As discussed in A007916, the expansion of a positive integer into a tower of NPPs is unique and always possible. 65536=2^2^2^2 is the smallest number that requires a tower of height more than 3.
%H David A. Corneth, <a href="/A277562/b277562.txt">Table of n, a(n) for n = 1..10025</a> (terms <= 10^16)
%H R. K. Guy and J. L. Selfridge, <a href="http://www.jstor.org/stable/2319392">The nesting and roosting habits of the laddered parenthesis</a>, Amer. Math. Monthly 80 (8) (1973), 868-876.
%H R. K. Guy and J. L. Selfridge, <a href="/A003018/a003018.pdf">The nesting and roosting habits of the laddered parenthesis</a> (annotated cached copy)
%e 16 = 2^2^2 81 = 3^2^2 256 = 2^2^3 512 = 2^3^2
%e 625 = 5^2^2 1296 = 6^2^2 2401 = 7^2^2 6561 = 3^2^3
%e 10000 = 10^2^2 14641 = 11^2^2 19683 = 3^3^2 20736 = 12^2^2
%e 28561 = 13^2^2 38416 = 14^2^2 50625 = 15^2^2
%e 65536 = 2^2^2^2 83521 = 17^2^2 104976 = 18^2^2 130321 = 19^2^2
%e 160000 = 20^2^2 194481 = 21^2^2 234256 = 22^2^2 279841 = 23^2^2
%e 331776 = 24^2^2 390625 = 5^2^3 456976 = 26^2^2 614656 = 28^2^2
%e 707281 = 29^2^2 810000 = 30^2^2 923521 = 31^2^2 1185921 = 33^2^2
%e 1336336 = 34^2^2 1500625 = 35^2^2 1679616 = 6^2^3 1874161 = 37^2^2
%e 1953125 = 5^3^2 2085136 = 38^2^2 2313441 = 39^2^2 2560000 = 40^2^2
%e 2825761 = 41^2^2 3111696 = 42^2^2 3418801 = 43^2^2 3748096 = 44^2^2
%e 4100625 = 45^2^2 4477456 = 46^2^2 4879681 = 47^2^2 5308416 = 48^2^2
%e 5764801 = 7^2^3 6250000 = 50^2^2 6765201 = 51^2^2 7311616 = 52^2^2
%e 7890481 = 53^2^2 8503056 = 54^2^2 9150625 = 55^2^2 9834496 = 56^2^2
%t radicalQ[1]:=False;
%t radicalQ[n_]:=SameQ[GCD@@FactorInteger[n][[All,2]],1];
%t hyperfactor[1]:={};
%t hyperfactor[n_?radicalQ]:={n};
%t hyperfactor[n_]:=With[{g=GCD@@FactorInteger[n][[All,2]]},Prepend[hyperfactor[g],Product[Apply[Power[#1,#2/g]&,r],{r,FactorInteger[n]}]]];
%t Select[Range[10^6],Length[hyperfactor[#]]>2&]
%Y Cf. A007916, A001597, A164336, A164337, A106490 (Quetian Superfactorization).
%K nonn
%O 1,1
%A _Gus Wiseman_, Oct 19 2016
%E Edited by _N. J. A. Sloane_, Nov 09 2016
%E Offset changed to 1 by _David A. Corneth_, Apr 30 2024