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.

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

Offset

1,1

Comments

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.

Links

David A. Corneth, Table of n, a(n) for n = 1..10025 (terms <= 10^16)

R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.

R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis (annotated cached copy)

Example

       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

Mathematica

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&]

Crossrefs

Cf. A007916, A001597, A164336, A164337, A106490 (Quetian Superfactorization).

Sequence in context: A250362 A217261 A372405 * A217709 A257854 A017672

Adjacent sequences: A277559 A277560 A277561 * A277563 A277564 A277565

Keyword

nonn

Author

Gus Wiseman, Oct 19 2016

Extensions

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

Offset changed to 1 by David A. Corneth, Apr 30 2024

Status

approved