A029935 a(n) = Sum_{d divides n} phi(d)*phi(n/d).

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• mult
• nonn

Comments

Dirichlet convolution of A000010 with itself. - R. J. Mathar, Aug 28 2015

Links

• Gheorghe Coserea, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000005, A000010, A001620, A005117, A018804, A029936, A143258.

Row sums of A159937.

Programs

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 *)

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

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