OFFSET
1,2
COMMENTS
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).
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)
FORMULA
Multiplicative with a(p^e) = p^(e + 1 - (e mod 2)) + 1.
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)).
From Vaclav Kotesovec, Sep 05 2023: (Start)
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)).
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)).
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(2) = Product_{p prime} (1 - 4/p^2 + 2/p^3 + 3/p^4 - 2/p^5) = 0.17432153313226756485612314112586411632220602294650993976966957787608316...,
f'(2) = f(2) * Sum_{p prime} 11 * log(p) / (p^2 + p - 2) = f(2) * 5.12969275236278527949034734003948649118572887258486718244613616120875581...
and gamma is the Euler-Mascheroni constant A001620. (End)
MATHEMATICA
f[p_, e_] := p^(e + 1 - Mod[e, 2]) + 1; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100]
PROG
(PARI) a(n) = {my(f = factor(n)); prod(i=1, #f~, f[i, 1]^(f[i, 2] + 1 - f[i, 2]%2) + 1); }
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Sep 05 2023
STATUS
approved