%I #42 Dec 06 2022 07:57:16

%S 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,

%T 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,

%U 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

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

%C Multiplicative because A003557 is. - _Andrew Howroyd_, Jul 27 2018

%H Antti Karttunen, <a href="/A300717/b300717.txt">Table of n, a(n) for n = 1..65537</a>

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

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

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

%F 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

%F Dirichlet convolution with A003958 equals A000010. - _Werner Schulte_, Sep 28 2018

%F a(n) = Sum_{d|n} mu(d)*phi(d)*phi(n/d). - _Ridouane Oudra_, Nov 18 2019

%F Dirichlet convolution of A000010 and A097945. - _R. J. Mathar_, Jun 02 2020

%F From _Richard L. Ollerton_, May 07 2021: (Start)

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

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

%p with(numtheory): A003557 := n -> n/ilcm(op(numtheory[factorset](n))):

%p seq(add(mobius(d)*A003557(n/d), d in divisors(n)), n=1..100); # _Ridouane Oudra_, Nov 18 2019

%t Table[DivisorSum[n, MoebiusMu[#] EulerPhi[#] EulerPhi[n/#] &], {n, 108}] (* _Michael De Vlieger_, Nov 18 2019 *)

%t 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 *)

%o (PARI)

%o A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); }; \\ From A003557

%o A300717(n) = sumdiv(n,d,moebius(n/d)*A003557(d));

%o (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

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

%K nonn,mult

%O 1,8

%A _Antti Karttunen_, Mar 11 2018