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A029935 a(n) = Sum_{d divides n} phi(d)*phi(n/d). 37
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Dirichlet convolution of A000010 with itself. - R. J. Mathar, Aug 28 2015
LINKS
FORMULA
From Vladeta Jovovic, Oct 30 2001: (Start)
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)
From Franklin T. Adams-Watters, Nov 19 2004: (Start)
Sum_{d|n} a(d) = A018804(n), Mobius transform of A018804.
Dirichlet g.f.: zeta(s-1)^2/zeta(s)^2. (End)
Equals row sums of triangle A143258. - Gary W. Adamson, Aug 02 2008
a(n) <= A000010(n) * A000005(n), with equality iff n = A005117(k) for some k. - Gheorghe Coserea, Oct 23 2016
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
MAPLE
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;
MATHEMATICA
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 *)
PROG
(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));
};
vector(54, n, a(n)) \\ Gheorghe Coserea, Oct 23 2016
(PARI) a(n) = sumdiv(n, d, eulerphi(d)*eulerphi(n/d)); \\ Michel Marcus, Oct 23 2016
CROSSREFS
Row sums of A159937.
Sequence in context: A345426 A061884 A286002 * A317625 A308447 A350506
KEYWORD
mult,nonn
AUTHOR
STATUS
approved

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Last modified March 29 04:59 EDT 2024. Contains 371264 sequences. (Running on oeis4.)