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%I #7 May 02 2024 10:06:39
%S 0,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,2,1,2,1,1,1,1,1,1,1,1,1,2,1,1,1,1,
%T 1,1,1,2,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,
%U 1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,2,1,1,2
%N a(n) = log_2(A372466(n) + 1).
%H Amiram Eldar, <a href="/A372467/b372467.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = log_2(A051903(A036537(n)) + 1).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (d(1) + Sum_{k>=2} (k * (d(k) - d(k-1))) / A327839 = 1.12132865776925625956..., where d(k) = Product_{p prime} (1 - 1/p + Sum_{i=1..k} (1/p^(2^i-1)-1/p^(2^i))).
%t pow2Q[n_] := n == 2^IntegerExponent[n, 2]; f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, pow2Q[# + 1] &], IntegerExponent[Max @@ e + 1, 2], Nothing]]; f[1] = 0; Array[f, 150]
%o (PARI) ispow2(n) = n >> valuation(n, 2) == 1;
%o lista(kmax) = {my(e); print1(0, ", "); for(k = 2, kmax, e = factor(k)[, 2]; if(ispow2(vecprod(apply(x -> x + 1, e))), print1(valuation(vecmax(e) + 1, 2), ", "))); }
%Y Cf. A036537, A051903, A327839, A369934, A372466.
%K nonn,easy
%O 1,7
%A _Amiram Eldar_, May 02 2024
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