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A368470 a(n) is the number of exponentially odd divisors of the largest unitary divisor of n that is an exponentially odd number (A268335).

%I A368470 #11 Dec 27 2023 01:21:36
%S A368470 1,2,2,1,2,4,2,3,1,4,2,2,2,4,4,1,2,2,2,2,4,4,2,6,1,4,3,2,2,8,2,4,4,4,
%T A368470 4,1,2,4,4,6,2,8,2,2,2,4,2,2,1,2,4,2,2,6,4,6,4,4,2,4,2,4,2,1,4,8,2,2,
%U A368470 4,8,2,3,2,4,2,2,4,8,2,2,1,4,2,4,4,4,4
%N A368470 a(n) is the number of exponentially odd divisors of the largest unitary divisor of n that is an exponentially odd number (A268335).
%H A368470 Amiram Eldar, <a href="/A368470/b368470.txt">Table of n, a(n) for n = 1..10000</a>
%H A368470 Vaclav Kotesovec, <a href="/A368470/a368470.jpg">Graph - the asymptotic ratio (100000 terms)</a>
%F A368470 a(n) = A033634(A350389(n)).
%F A368470 Multiplicative with a(p^e) = (e+3)/2 if e is odd and 1 otherwise.
%F A368470 a(n) >= 1, with equality if and only if n is a square (A000290).
%F A368470 a(n) <= A000005(n), with equality if and only if n is squarefree (A005117).
%F A368470 Dirichlet g.f.: zeta(2*s)^2 * Product_{p prime} (1 + 2/p^s - 1/p^(2*s) - 1/p^(3*s)).
%F A368470 From _Vaclav Kotesovec_, Dec 26 2023: (Start)
%F A368470 Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s + p^s/(1 + p^s)^2).
%F A368470 Let f(s) = Product_{p prime} (1 - 1/p^s + p^s/(1 + p^s)^2).
%F A368470 Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
%F A368470 f(1) = Product_{p prime} (1 - (2*p+1) / (p*(p+1)^2)) = 0.528940778823659679133966695786017426052491935740673837882972347697...,
%F A368470 f'(1) = f(1) * Sum_{p prime} (4*p^2 + 3*p + 1) * log(p) / (p^4 + 3*p^3 + p^2 - 2*p - 1) = f(1) * 1.36109933267802415215189866467122940932493907539386280428818...
%F A368470 and gamma is the Euler-Mascheroni constant A001620. (End)
%t A368470 f[p_, e_] := If[OddQ[e], (e + 3)/2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A368470 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2]%2, (f[i,2]+3)/2, 1));}
%Y A368470 Cf. A000005, A000290, A005117, A033634, A268335, A350389, A368471.
%K A368470 nonn,easy,mult
%O A368470 1,2
%A A368470 _Amiram Eldar_, Dec 26 2023