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A029935
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a(n) = Sum_{d divides n} phi(d)*phi(n/d).
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37
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1, 2, 4, 5, 8, 8, 12, 12, 16, 16, 20, 20, 24, 24, 32, 28, 32, 32, 36, 40, 48, 40, 44, 48, 56, 48, 60, 60, 56, 64, 60, 64, 80, 64, 96, 80, 72, 72, 96, 96, 80, 96, 84, 100, 128, 88, 92, 112, 120, 112, 128, 120, 104, 120
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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Sum_{k=1..n} phi(gcd(n, k)).
Multiplicative with a(p^e) = (e+1)*(p^e - p^(e - 1)) - (e - 1)*(p^(e - 1) - p^(e - 2)). (End)
Dirichlet g.f.: zeta(s-1)^2/zeta(s)^2. (End)
Sum_{k=1..n} a(k) ~ 9*n^2 * ((2*log(n) + 4*gamma - 1)/Pi^4 - 24*Zeta'(2)/Pi^6), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 31 2019
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MAPLE
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with(numtheory): A029935 := proc(n) local i, j; j := 0; for i in divisors(n) do j := j+phi(i)*phi(n/i); od; j; end;
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MATHEMATICA
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A029935[n_]:=DivisorSum[n, EulerPhi[#]*EulerPhi[n/#]&]; Array[A029935, 50]
f[p_, e_] := (e+1)*(p^e - p^(e-1)) - (e-1)*(p^(e-1) - p^(e-2)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 26 2023 *)
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PROG
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(PARI)
a(n) = {
my(f = factor(n), fsz = matsize(f)[1],
g = prod(k=1, fsz, f[k, 1]),
h = prod(k=1, fsz, sqr(f[k, 1]-1)*f[k, 2] + sqr(f[k, 1])-1));
return(h*n\sqr(g));
};
(PARI) a(n) = sumdiv(n, d, eulerphi(d)*eulerphi(n/d)); \\ Michel Marcus, Oct 23 2016
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CROSSREFS
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KEYWORD
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mult,nonn
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AUTHOR
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STATUS
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approved
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