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A386262 a(n) = A051903(A051903(n)) for n >= 2, a(1) = 0.

%I A386262 #7 Jul 20 2025 15:39:30
%S A386262 0,0,0,1,0,0,0,1,1,0,0,1,0,0,0,2,0,1,0,1,0,0,0,1,1,0,1,1,0,0,0,1,0,0,
%T A386262 0,1,0,0,0,1,0,0,0,1,1,0,0,2,1,1,0,1,0,1,0,1,0,0,0,1,0,0,1,1,0,0,0,1,
%U A386262 0,0,0,1,0,0,1,1,0,0,0,2,2,0,0,1,0,0,0
%N A386262 a(n) = A051903(A051903(n)) for n >= 2, a(1) = 0.
%C A386262 The first occurrence of k = 1, 2, ... is at n = 2^(2^k) = A001146(k).
%C A386262 If n is an exponentially squarefree number (A209061) then a(n) <= 1. The converse is not necessarily true, with n = 2592 = 2^5 * 3^4 being the least counterexample.
%C A386262 The asymptotic density of the occurrences of 0 in this sequence is 1/zeta(2) = 6/Pi^2 (A059956).
%C A386262 The asymptotic density of the occurrences of 1 in this sequence is Sum_{k squarefree > 1} (1/zeta(k+1) - 1/zeta(k)) = 0.348423339572619656701... .
%H A386262 Amiram Eldar, <a href="/A386262/b386262.txt">Table of n, a(n) for n = 1..10000</a>
%F A386262 a(n) = 0 if and only if n is squarefree (A005117).
%F A386262 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=2} A051903(k) * (1/zeta(k+1)-1/zeta(k)) = 0.43779421197744649258... .
%t A386262 f[n_] := Max[FactorInteger[n][[;; , 2]]]; f[1] = 0; a[n_] := f[f[n]]; a[1] = 0; Array[a, 100]
%o A386262 (PARI) f(n) = if(n == 1, 0, vecmax(factor(n)[,2]));
%o A386262 a(n) = if(n == 1, 0, f(f(n)));
%Y A386262 Cf. A005117, A001146, A051903, A059956, A209061.
%Y A386262 Similar sequences: A010553, A010554, A051027, A055033, A058699, A068346, A132090, A178601, A181776, A284908, A386261.
%K A386262 nonn,easy
%O A386262 1,16
%A A386262 _Amiram Eldar_, Jul 17 2025