A304481 Turn the power-tower for n upside-down.

1, 2, 3, 4, 5, 6, 7, 9, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 32, 26, 27, 28, 29, 30, 31, 25, 33, 34, 35, 64, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 128, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 36, 65, 66, 67

Offset

1,2

Comments

This is an involution of the positive integers.

The power-tower for n is defined as follows. 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. Then a(n) = c(x_k)^...^c(x_3)^c(x_2)^c(x_1).

Links

Robert Israel, Table of n, a(n) for n = 1..10000

N. J. A. Sloane, Maple programs for A007916, A278028, A278029, A052409, A089723, A277564

Example

The power tower of 81 is 3^2^2, which turned upside-down is 2^2^3 = 256, so a(81) = 256.

Maple

f:= proc(n, r) local F, a, y;
     if n = 1 then return 1 fi;
     F:= ifactors(n)[2];
     y:= igcd(seq(t[2], t=F));
     if y = 1 then return n^r fi;
     a:= mul(t[1]^(t[2]/y), t=F);
     procname(y, a^r)
end proc:
seq(f(n, 1), n=1..100); # Robert Israel, May 13 2018

Mathematica

tow[n_]:=If[n==1, {}, With[{g=GCD@@FactorInteger[n][[All, 2]]}, If[g===1, {n}, Prepend[tow[g], n^(1/g)]]]];
Table[Power@@Reverse[tow[n]], {n, 100}]

Crossrefs

Cf. A052409, A052410, A007916, A089723, A164337, A277562, A277564, A278028, A288636, A289023, A294336, A294337, A304491, A304492, A304495.

Sequence in context: A021085 A031006 A031978 * A222254 A235489 A065306

Adjacent sequences: A304478 A304479 A304480 * A304482 A304483 A304484

Keyword

nonn,look

Author

Gus Wiseman, May 13 2018

Status

approved