A300717 Möbius transform of A003557, n divided by its largest squarefree divisor.

1, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 6, 0, 0, 0, 0, 8, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 6

Offset: 1

Author

Antti Karttunen, Mar 11 2018

Keywords

• nonn
• mult

Comments

Multiplicative because A003557 is. - Andrew Howroyd, Jul 27 2018

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A000010, A008683, A003557, A300718, A300719, A003958.

Programs

Maple

with(numtheory): A003557 := n -> n/ilcm(op(numtheory[factorset](n))):
seq(add(mobius(d)*A003557(n/d), d in divisors(n)), n=1..100); # Ridouane Oudra, Nov 18 2019

Mathematica

Table[DivisorSum[n, MoebiusMu[#] EulerPhi[#] EulerPhi[n/#] &], {n, 108}] (* Michael De Vlieger, Nov 18 2019 *)
f[p_, e_] := If[e == 1, 0, (p - 1)*p^(e - 2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 06 2022 *)

PARI

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); }; \\ From A003557
A300717(n) = sumdiv(n,d,moebius(n/d)*A003557(d));

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, 0, (f[i,1] - 1)*f[i,1]^(f[i,2] - 2)));} \\ Amiram Eldar, Dec 06 2022

Formula

a(n) = Sum_{d|n} A008683(n/d)*A003557(d).

a(n) = A000010(n) - A300718(n).

a(n) = A003557(n) - A300719(n).

Multiplicative with a(p) = 0 and a(p^e) = (p-1)*p^(e-2) for prime p and e>1. - Werner Schulte, Sep 27 2018

Dirichlet convolution with A003958 equals A000010. - Werner Schulte, Sep 28 2018

a(n) = Sum_{d|n} mu(d)*phi(d)*phi(n/d). - Ridouane Oudra, Nov 18 2019

Dirichlet convolution of A000010 and A097945. - R. J. Mathar, Jun 02 2020

From Richard L. Ollerton, May 07 2021: (Start)

a(n) = Sum_{k=1..n} phi(gcd(n,k))*mu(gcd(n,k)).

a(n) = Sum_{k=1..n} phi(gcd(n,k))*mu(n/gcd(n,k)). (End)