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A277562
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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.
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14
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16, 81, 256, 512, 625, 1296, 2401, 6561, 10000, 14641, 19683, 20736, 28561, 38416, 50625, 65536, 83521, 104976, 130321, 160000, 194481, 234256, 279841, 331776, 390625, 456976, 614656, 707281, 810000, 923521, 1185921, 1336336, 1500625, 1679616, 1874161, 1953125, 2085136, 2313441, 2560000, 2825761, 3111696, 3418801
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OFFSET
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3,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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16 = 2^2^2 81 = 3^2^2 256 = 2^2^3 512 = 2^3^2
625 = 5^2^2 1296 = 6^2^2 2401 = 7^2^2 6561 = 3^2^3
10000 = 10^2^2 14641 = 11^2^2 19683 = 3^3^2 20736 = 12^2^2
28561 = 13^2^2 38416 = 14^2^2 50625 = 15^2^2
65536 = 2^2^2^2 83521 = 17^2^2 104976 = 18^2^2 130321 = 19^2^2
160000 = 20^2^2 194481 = 21^2^2 234256 = 22^2^2 279841 = 23^2^2
331776 = 24^2^2 390625 = 5^2^3 456976 = 26^2^2 614656 = 28^2^2
707281 = 29^2^2 810000 = 30^2^2 923521 = 31^2^2 1185921 = 33^2^2
1336336 = 34^2^2 1500625 = 35^2^2 1679616 = 6^2^3 1874161 = 37^2^2
1953125 = 5^3^2 2085136 = 38^2^2 2313441 = 39^2^2 2560000 = 40^2^2
2825761 = 41^2^2 3111696 = 42^2^2 3418801 = 43^2^2 3748096 = 44^2^2
4100625 = 45^2^2 4477456 = 46^2^2 4879681 = 47^2^2 5308416 = 48^2^2
5764801 = 7^2^3 6250000 = 50^2^2 6765201 = 51^2^2 7311616 = 52^2^2
7890481 = 53^2^2 8503056 = 54^2^2 9150625 = 55^2^2 9834496 = 56^2^2
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MATHEMATICA
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radicalQ[1]:=False;
radicalQ[n_]:=SameQ[GCD@@FactorInteger[n][[All, 2]], 1];
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]}]]];
Select[Range[10^6], Length[hyperfactor[#]]>2&]
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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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