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A368977
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The number of bi-unitary divisors of n that are exponentially odd numbers (A268335).
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3
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1, 2, 2, 1, 2, 4, 2, 3, 1, 4, 2, 2, 2, 4, 4, 3, 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, 6, 1, 2, 4, 2, 2, 6, 4, 6, 4, 4, 2, 4, 2, 4, 2, 3, 4, 8, 2, 2, 4, 8, 2, 3, 2, 4, 2, 2, 4, 8, 2, 6, 3, 4, 2, 4, 4, 4, 4
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OFFSET
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1,2
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LINKS
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FORMULA
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Multiplicative with a(p^e) = (e+3)/2 if e is odd, and 2*floor(e/4)+1 if e is even.
a(n) >= 1, with equality if and only if n is in A062503.
a(n) <= A000005(n), with equality if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - (1 - p^s + 2*p^(2*s)) / (p^s*(1 + p^s)*(1 + p^(2*s)))).
Let f(s) = Product_{p prime} (1 - (1 - p^s + 2*p^(2*s)) / (p^s*(1 + p^s)*(1 + p^(2*s)))).
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 - (1 - p + 2*p^2) / (p*(1 + p)*(1 + p^2))) = 0.5715031234451924252215041182933420817059774181158824297150124265420835...,
f'(1) = f(1) * Sum_{p prime} (4*p^5 - p^4 + 2*p^3 + 2*p + 1) * log(p) / (p^7 + 2*p^6 + p^5 + 3*p^4 + p^3 + p - 1) = f(1) * 1.1422556395248477875508983912036578244050011522937179465478688905880430...
and gamma is the Euler-Mascheroni constant A001620. (End)
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MATHEMATICA
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f[p_, e_] := If[OddQ[e], (e+3)/2, 2*Floor[e/4]+1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
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PROG
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(PARI) a(n) = vecprod(apply(x -> if(x%2, (x+3)/2, 2*(x\4)+1), factor(n)[, 2]));
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + X - X^2 + 2*X^3 - X^4)/(1 - X - X^4 + X^5))[n], ", ")) \\ Vaclav Kotesovec, Jan 11 2024
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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