A352080 a(n) is the number of times that the square root operation must be applied to n in order to reach an irrational number.

1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 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, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1

Offset

2,3

Comments

a(1) is undefined because 1^(1/2^k) = 1 for all k.

Column a(n)-1 has the first nonunit term in row n of A352780. - Peter Munn, Nov 15 2022

Links

Table of n, a(n) for n=2..108.

Formula

a(n) is the minimum k such that n^(1/2^k) is irrational.

a(n) = A007814(A052409(n)) + 1. - Amiram Eldar, Mar 03 2022

a(n) = A001511(A267116(n)). - Peter Munn, Nov 15 2022

Example

a(2) = 1 because sqrt(2) is irrational.
a(16) = 3 because sqrt(16) = 16^(1/2) = 4, sqrt(sqrt(16)) = 16^(1/4) = 2, but sqrt(sqrt(sqrt(16))) = 16^(1/8) = sqrt(2), which is irrational.

Mathematica

a[n_] := IntegerExponent[GCD @@ FactorInteger[n][[;; , 2]], 2] + 1; Array[a, 100, 2] (* Amiram Eldar, Mar 03 2022 *)

Prog

(PARI) a(n) = if (!issquare(n, &n), 1, a(n)+1); \\ Michel Marcus, Mar 03 2022

Crossrefs

Cf. A000290 (squares), A010052.

See the formula section for the relationships with A001511, A007814, A052409, A267116.

Cf. also A000037 (indices of 1's), A030140 (indices of 2's).

Cf. A352780.

Sequence in context: A074064 A275215 A304886 * A295632 A139549 A216915

Adjacent sequences: A352077 A352078 A352079 * A352081 A352082 A352083

Keyword

nonn

Author

Ryan Jean, Mar 02 2022

Status

approved