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A365173 The number of divisors d of n such that gcd(d, n/d) is an exponentially odd number (A268335).

%I A365173 #18 Apr 27 2025 00:45:25
%S A365173 1,2,2,3,2,4,2,4,3,4,2,6,2,4,4,4,2,6,2,6,4,4,2,8,3,4,4,6,2,8,2,4,4,4,
%T A365173 4,9,2,4,4,8,2,8,2,6,6,4,2,8,3,6,4,6,2,8,4,8,4,4,2,12,2,4,6,5,4,8,2,6,
%U A365173 4,8,2,12,2,4,6,6,4,8,2,8,4,4,2,12,4,4
%N A365173 The number of divisors d of n such that gcd(d, n/d) is an exponentially odd number (A268335).
%C A365173 First differs from A252505 at n = 64.
%C A365173 The sum of these divisors is A365174(n).
%H A365173 Amiram Eldar, <a href="/A365173/b365173.txt">Table of n, a(n) for n = 1..10000</a>
%H A365173 Vaclav Kotesovec, <a href="/A365173/a365173.jpg">Graph - the asymptotic ratio (100000 terms)</a>
%F A365173 Multiplicative with a(p^e) = floor((e+5)/4) + floor((e+6)/4) = A004524(e+5).
%F A365173 a(n) <= A000005(n), with equality if and only if n is not a biquadrateful number (A046101).
%F A365173 a(n) >= A034444(n), with equality if and only if n is squarefree (A005117).
%F A365173 a(n) == 1 (mod 2) if and only if n is a square of an exponentially odd number (i.e., a number whose prime factorization include only exponents e such that e == 2 (mod 4)).
%F A365173 From _Vaclav Kotesovec_, Jan 20 2024: (Start)
%F A365173 Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/(p^(2*s)*(1 + p^(2*s)))).
%F A365173 Let f(s) = Product_{p prime} (1 - 1/(p^(2*s)*(1 + p^(2*s)))).
%F A365173 Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
%F A365173 f(1) = Product_{p prime} (1 - 1/(p^2*(1 + p^2))) = 0.937494282731300250789438325050116436995101826036120273493270589183132928...,
%F A365173 f'(1) = f(1) * Sum_{p prime} (4*p^2 + 2) * log(p) / (p^6 + 2*p^4 - 1) = f(1) * 0.192452062257404507109731932640803706644036700262364333369815000973104583...
%F A365173 and gamma is the Euler-Mascheroni constant A001620. (End)
%t A365173 f[p_, e_] := Floor[(e + 5)/4] + Floor[(e + 6)/4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A365173 (PARI) a(n) = vecprod(apply(x -> (x+5)\4 + (x+6)\4, factor(n)[, 2]));
%o A365173 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + X^2 - X^4)/((1 - X)^2*(1 + X^2)))[n], ", ")) \\ _Vaclav Kotesovec_, Jan 20 2024
%Y A365173 Cf. A004524, A046101, A005117, A252505, A268335, A350390, A325837, A365171, A365174.
%K A365173 nonn,easy,mult
%O A365173 1,2
%A A365173 _Amiram Eldar_, Aug 25 2023