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A365348
The number of divisors of the smallest exponentially odd number divisible by n.
3
1, 2, 2, 4, 2, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 6, 2, 8, 2, 8, 4, 4, 2, 8, 4, 4, 4, 8, 2, 8, 2, 6, 4, 4, 4, 16, 2, 4, 4, 8, 2, 8, 2, 8, 8, 4, 2, 12, 4, 8, 4, 8, 2, 8, 4, 8, 4, 4, 2, 16, 2, 4, 8, 8, 4, 8, 2, 8, 4, 8, 2, 16, 2, 4, 8, 8, 4, 8, 2, 12, 6, 4, 2, 16, 4
OFFSET
1,2
COMMENTS
The sum of these divisors is A365349(n).
FORMULA
a(n) = A000005(A356191(n)).
Multiplicative with a(p^e) = e + 2 - (e mod 2).
Dirichlet g.f.: zeta(s)^2 * zeta(2*s) * Product_{p prime} (1 - 2/p^(3*s) + 1/p^(4*s)).
From Vaclav Kotesovec, Sep 05 2023: (Start)
Let f(s) = Product_{p prime} (1 - 2/p^(3*s) + 1/p^(4*s)).
Sum_{k=1..n} a(k) ~ f(1) * Pi^2 * n / 6 * (log(n) + 2*gamma - 1 + 12*zeta'(2)/Pi^2 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 2/p^3 + 1/p^4) = 0.74469549790606742043912387159445432811796913290492411186307181370150975026...
f'(1) = f(1) * Sum_{p prime} 2*(3*p - 2) * log(p) / (1 - 2*p + p^4) = f(1) * 0.75575434641494973924789411019492794958528241212857430737760075121773728338...
and gamma is the Euler-Mascheroni constant A001620. (End)
MATHEMATICA
f[p_, e_] := e + 2 - Mod[e, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = vecprod(apply(x -> x + 2 - x%2, factor(n)[, 2]));
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Sep 02 2023
STATUS
approved