A304495 Decapitate the power-tower for n, i.e., remove the last (deepest) exponent.

0, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1

Offset

1,4

Comments

a(1) = 0 by convention.

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_1)^c(x_2)^c(x_3)^...^c(x_{k-1}).

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

a(m) <> 1 if m is a perfect power (A001597). - Michel Marcus, Jul 23 2018

Example

We have 64 = 2^6, so a(64) = 2.
We have 216 = 6^3, so a(216) = 6.
We have 256 = 2^2^3, so a(256) = 2^2 = 4.

Mathematica

tow[n_]:=If[n==1, {}, With[{g=GCD@@FactorInteger[n][[All, 2]]}, If[g===1, {n}, Prepend[tow[g], n^(1/g)]]]];
Table[If[n==1, 0, Power@@Most[tow[n]]], {n, 100}]

Prog

(PARI) A304495(n) = if(1==n, 0, my(e, r, tow = List([])); while((e = ispower(n, , &r)) > 1, listput(tow, r); n = e; ); n = 1; while(length(tow)>0, e = tow[#tow]; listpop(tow); n = e^n; ); (n)); \\ Antti Karttunen, Jul 23 2018

Crossrefs

Cf. A052409, A052410, A001597, A007916, A089723, A164337, A277562, A277564, A278028, A288636, A289023, A294336, A294337, A304481, A304491, A304492.

Sequence in context: A131256 A362414 A245562 * A175069 A245563 A356917

Adjacent sequences: A304492 A304493 A304494 * A304496 A304497 A304498

Keyword

nonn

Author

Gus Wiseman, May 13 2018

Extensions

Name edited and more terms from Antti Karttunen, Jul 23 2018

Status

approved