This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A383006 #7 Apr 12 2025 09:41:46
%S A383006 0,2,3,0,4,0,0,5,2,0,6,3,0,2,2,0,0,7,4,0,2,3,3,0,0,8,5,0,2,0,0,3,4,4,
%T A383006 0,2,2,9,0,6,0,3,0,2,0,4,5,0,5,2,0,3,2,3,10,0,0,7,2,0,4,0,0,3,0,2,2,0,
%U A383006 5,6,0,6,2,3,0,4,3,4,0,11,2,0,0,0,8,3,0
%N A383006 Exponent of the highest power of 2 dividing the n-th powerful number.
%H A383006 Amiram Eldar, <a href="/A383006/b383006.txt">Table of n, a(n) for n = 1..10000</a>
%H A383006 <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.
%F A383006 a(n) = A007814(A001694(n)).
%F A383006 a(A363189(n)) = 0.
%F A383006 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (11 + 6*sqrt(2))/7 = 2.7836116... .
%t A383006 IntegerExponent[Select[Range[5000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &], 2]
%o A383006 (PARI) list(lim) = for(k = 1, lim, if(ispowerful(k), print1(valuation(k, 2), ", ")));
%Y A383006 Cf. A001694, A007814, A363189, A373549, A383007.
%K A383006 nonn,easy
%O A383006 1,2
%A A383006 _Amiram Eldar_, Apr 12 2025