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

Author

Ryan Jean, Mar 02 2022

Keywords

• nonn

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

Examples

			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.
		

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.

Programs

Mathematica

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

PARI

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

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