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A277562 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.

%I A277562 #27 May 01 2024 01:45:54
%S A277562 16,81,256,512,625,1296,2401,6561,10000,14641,19683,20736,28561,38416,
%T A277562 50625,65536,83521,104976,130321,160000,194481,234256,279841,331776,
%U A277562 390625,456976,614656,707281,810000,923521,1185921,1336336,1500625,1679616,1874161,1953125,2085136,2313441,2560000,2825761,3111696,3418801
%N A277562 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 A277562 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 A277562 David A. Corneth, <a href="/A277562/b277562.txt">Table of n, a(n) for n = 1..10025</a> (terms <= 10^16)
%H A277562 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 A277562 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 A277562        16 = 2^2^2        81 = 3^2^2       256 = 2^2^3       512 = 2^3^2
%e A277562       625 = 5^2^2      1296 = 6^2^2      2401 = 7^2^2      6561 = 3^2^3
%e A277562     10000 = 10^2^2    14641 = 11^2^2    19683 = 3^3^2     20736 = 12^2^2
%e A277562     28561 = 13^2^2    38416 = 14^2^2    50625 = 15^2^2
%e A277562     65536 = 2^2^2^2   83521 = 17^2^2   104976 = 18^2^2   130321 = 19^2^2
%e A277562    160000 = 20^2^2   194481 = 21^2^2   234256 = 22^2^2   279841 = 23^2^2
%e A277562    331776 = 24^2^2   390625 = 5^2^3    456976 = 26^2^2   614656 = 28^2^2
%e A277562    707281 = 29^2^2   810000 = 30^2^2   923521 = 31^2^2  1185921 = 33^2^2
%e A277562   1336336 = 34^2^2  1500625 = 35^2^2  1679616 = 6^2^3   1874161 = 37^2^2
%e A277562   1953125 = 5^3^2   2085136 = 38^2^2  2313441 = 39^2^2  2560000 = 40^2^2
%e A277562   2825761 = 41^2^2  3111696 = 42^2^2  3418801 = 43^2^2  3748096 = 44^2^2
%e A277562   4100625 = 45^2^2  4477456 = 46^2^2  4879681 = 47^2^2  5308416 = 48^2^2
%e A277562   5764801 = 7^2^3   6250000 = 50^2^2  6765201 = 51^2^2  7311616 = 52^2^2
%e A277562   7890481 = 53^2^2  8503056 = 54^2^2  9150625 = 55^2^2  9834496 = 56^2^2
%t A277562 radicalQ[1]:=False;
%t A277562 radicalQ[n_]:=SameQ[GCD@@FactorInteger[n][[All,2]],1];
%t A277562 hyperfactor[1]:={};
%t A277562 hyperfactor[n_?radicalQ]:={n};
%t A277562 hyperfactor[n_]:=With[{g=GCD@@FactorInteger[n][[All,2]]},Prepend[hyperfactor[g],Product[Apply[Power[#1,#2/g]&,r],{r,FactorInteger[n]}]]];
%t A277562 Select[Range[10^6],Length[hyperfactor[#]]>2&]
%Y A277562 Cf. A007916, A001597, A164336, A164337, A106490 (Quetian Superfactorization).
%K A277562 nonn
%O A277562 1,1
%A A277562 _Gus Wiseman_, Oct 19 2016
%E A277562 Edited by _N. J. A. Sloane_, Nov 09 2016
%E A277562 Offset changed to 1 by _David A. Corneth_, Apr 30 2024