%I #32 Nov 16 2022 08:55:19
%S 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,
%T 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,
%U 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
%N a(n) is the number of times that the square root operation must be applied to n in order to reach an irrational number.
%C a(1) is undefined because 1^(1/2^k) = 1 for all k.
%C Column a(n)-1 has the first nonunit term in row n of A352780. - _Peter Munn_, Nov 15 2022
%F a(n) is the minimum k such that n^(1/2^k) is irrational.
%F a(n) = A007814(A052409(n)) + 1. - _Amiram Eldar_, Mar 03 2022
%F a(n) = A001511(A267116(n)). - _Peter Munn_, Nov 15 2022
%e a(2) = 1 because sqrt(2) is irrational.
%e 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.
%t a[n_] := IntegerExponent[GCD @@ FactorInteger[n][[;; , 2]], 2] + 1; Array[a, 100, 2] (* _Amiram Eldar_, Mar 03 2022 *)
%o (PARI) a(n) = if (!issquare(n, &n), 1, a(n)+1); \\ _Michel Marcus_, Mar 03 2022
%Y Cf. A000290 (squares), A010052.
%Y See the formula section for the relationships with A001511, A007814, A052409, A267116.
%Y Cf. also A000037 (indices of 1's), A030140 (indices of 2's).
%Y Cf. A352780.
%K nonn
%O 2,3
%A _Ryan Jean_, Mar 02 2022