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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).
2
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, 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, 4, 8, 2, 3, 2, 4, 2, 2, 4, 8, 2, 2, 1, 4, 2, 4, 4, 4, 4
OFFSET
1,2
FORMULA
a(n) = A033634(A350389(n)).
Multiplicative with a(p^e) = (e+3)/2 if e is odd and 1 otherwise.
a(n) >= 1, with equality if and only if n is a square (A000290).
a(n) <= A000005(n), with equality if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(2*s)^2 * Product_{p prime} (1 + 2/p^s - 1/p^(2*s) - 1/p^(3*s)).
From Vaclav Kotesovec, Dec 26 2023: (Start)
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s + p^s/(1 + p^s)^2).
Let f(s) = Product_{p prime} (1 - 1/p^s + p^s/(1 + p^s)^2).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - (2*p+1) / (p*(p+1)^2)) = 0.528940778823659679133966695786017426052491935740673837882972347697...,
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...
and gamma is the Euler-Mascheroni constant A001620. (End)
MATHEMATICA
f[p_, e_] := If[OddQ[e], (e + 3)/2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2]%2, (f[i, 2]+3)/2, 1)); }
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Dec 26 2023
STATUS
approved