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A029935 a(n) = Sum phi(d)*phi(n/d); d divides n. 12
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

Sum_{d|n} a(d) = A018804(n), Mobius transform of A018804. - Franklin T. Adams-Watters, Nov 19 2004

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

LINKS

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

FORMULA

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)). - Vladeta Jovovic, Oct 30 2001

Dirichlet g.f.: zeta(s-1)^2/zeta(s)^2. - Franklin T. Adams-Watters, Nov 19 2004

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

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]

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

Cf. A029936. Row sums of A159937.

Sequence in context: A036694 A085624 A061884 * A123291 A099402 A248387

Adjacent sequences:  A029932 A029933 A029934 * A029936 A029937 A029938

KEYWORD

mult,nonn

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified April 30 08:33 EDT 2017. Contains 285645 sequences.