A304491 Last or deepest exponent in the power-tower for n.

1, 2, 3, 2, 5, 6, 7, 3, 2, 10, 11, 12, 13, 14, 15, 2, 17, 18, 19, 20, 21, 22, 23, 24, 2, 26, 3, 28, 29, 30, 31, 5, 33, 34, 35, 2, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 2, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 6, 65, 66, 67, 68, 69

Offset

1,2

Comments

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).

Links

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Formula

a(n) = A007916(A278028(n, A288636(n))).

Example

We have 16 = 2^2^2, so a(16) = 2.
We have 64 = 2^6, so a(64) = 6.
We have 81 = 3^2^2, so a(81) = 2.
We have 256 = 2^2^3, so a(256) = 3.

Mathematica

a[n_]:=If[n==1, 1, With[{g=GCD@@FactorInteger[n][[All, 2]]}, If[g==1, n, a[g]]]];
Array[a, 100]

Prog

(PARI) a(n)={my(t=n); while(t, n=t; t=ispower(t)); n} \\ Andrew Howroyd, Aug 26 2018

Crossrefs

Cf. A052409, A052410, A007916, A089723, A164337, A277562, A277564, A278028, A288636, A289023, A294336, A294337, A304481, A304492, A304495.

Sequence in context: A259445 A080979 A287620 * A277803 A062789 A066069

Adjacent sequences: A304488 A304489 A304490 * A304492 A304493 A304494

Keyword

nonn

Author

Gus Wiseman, May 13 2018

Status

approved