%I #12 Sep 06 2023 01:24:07

%S 1,3,4,9,6,12,8,9,28,18,12,36,14,24,24,33,18,84,20,54,32,36,24,36,126,

%T 42,28,72,30,72,32,33,48,54,48,252,38,60,56,54,42,96,44,108,168,72,48,

%U 132,344,378,72,126,54,84,72,72,80,90,60,216,62,96,224,129,84

%N The sum of unitary divisors of the smallest exponentially odd number that is divisible by n.

%C The number of unitary divisors of the smallest exponentially odd number that is divisible by n is the same as the number of unitary divisors of n, A034444(n).

%H Amiram Eldar, <a href="/A365480/b365480.txt">Table of n, a(n) for n = 1..10000</a>

%H Vaclav Kotesovec, <a href="/A365480/a365480.jpg">Graph - the asymptotic ratio (1000000 terms)</a>

%F a(n) = A034448(A356191(n)).

%F Multiplicative with a(p^e) = p^(e + 1 - (e mod 2)) + 1.

%F Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^(2*s-3) - 1/p^(2*s-2) - 1/p^(2*s-1) - 1/p^(3*s-3)).

%F From _Vaclav Kotesovec_, Sep 05 2023: (Start)

%F Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(2*s-2) * zeta(2*s-3) * Product_{p prime} (1 - p^(7-6*s) - p^(5-5*s) + p^(7-5*s) + 2*p^(4-4*s) + 2*p^(5-4*s) - p^(6-4*s) + p^(2-3*s) - p^(4-3*s) - p^(1-2*s) - 2*p^(2-2*s)).

%F Let f(s) = Product_{p prime} (1 - p^(7-6*s) - p^(5-5*s) + p^(7-5*s) + 2*p^(4-4*s) + 2*p^(5-4*s) - p^(6-4*s) + p^(2-3*s) - p^(4-3*s) - p^(1-2*s) - 2*p^(2-2*s)).

%F Sum_{k=1..n} a(k) ~ n^2 * Pi^4 * f(2) / 144 * (log(n) + 3*gamma - 1/2 + 18*zeta'(2)/Pi^2 + f'(2)/f(2)), where

%F f(2) = Product_{p prime} (1 - 4/p^2 + 2/p^3 + 3/p^4 - 2/p^5) = 0.17432153313226756485612314112586411632220602294650993976966957787608316...,

%F f'(2) = f(2) * Sum_{p prime} 11 * log(p) / (p^2 + p - 2) = f(2) * 5.12969275236278527949034734003948649118572887258486718244613616120875581...

%F and gamma is the Euler-Mascheroni constant A001620. (End)

%t f[p_, e_] := p^(e + 1 - Mod[e, 2]) + 1; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100]

%o (PARI) a(n) = {my(f = factor(n)); prod(i=1, #f~, f[i,1]^(f[i,2] + 1 - f[i,2]%2) + 1);}

%Y Cf. A034444, A034448, A356191, A365349, A365479, A365481.

%K nonn,easy,mult

%O 1,2

%A _Amiram Eldar_, Sep 05 2023