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A130107 Möbius transform of A063659.

%I A130107 #35 Feb 20 2022 14:37:58
%S A130107 1,1,2,1,4,2,6,3,5,4,10,2,12,6,8,6,16,5,18,4,12,10,22,6,19,12,16,6,28,
%T A130107 8,30,12,20,16,24,5,36,18,24,12,40,12,42,10,20,22,46,12,41,19,32,12,
%U A130107 52,16,40,18,36,28,58,8,60,30,30,24,48,20,66,16,44,24
%N A130107 Möbius transform of A063659.
%C A130107 Double inverse Möbius transform of A130107 = A001615: (1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, ...).
%H A130107 Enrique Pérez Herrero, <a href="/A130107/b130107.txt">Table of n, a(n) for n = 1..2000</a>
%H A130107 Steven R. Finch, <a href="/A001616/a001616.pdf">Primitive Cusp Forms</a>, April 27, 2009. [Cached copy, with permission of the author]
%F A130107 A054525 * A063659, (1, 2, 3, 3, 5, 6, 7, 6, 8, 10, ...).
%F A130107 Multiplicative with a(p^e) = p-1 if e=1,  a(p^e) = p^2-p-1 if e=2, a(p^e) = p^(e-3)*(p+1)*(p-1)^2. - _Enrique Pérez Herrero_, Apr 03 2014
%F A130107 Dirichlet g.f.: zeta(s-1) / (zeta(s) * zeta(2s)). - _Álvar Ibeas_, Mar 07 2015
%F A130107 Sum_{k=1..n} a(k) ~ 270*n^2 / Pi^6. - _Vaclav Kotesovec_, Jan 11 2019
%e A130107 G.f. = x + x^2 + 2*x^3 + x^4 + 4*x^5 + 2*x^6 + 6*x^7 + 3*x^8 + 5*x^9 + ...
%p A130107 with(numtheory): A130107 := proc(n) local dp, mtdp, d, p;
%p A130107 dp := n -> n*mul((1+1/p), p=factorset(n));
%p A130107 mtdp := n -> add(mobius(n/d)*dp(d), d=divisors(n));
%p A130107 add(mobius(n/d)*mtdp(d), d=divisors(n)) end:
%p A130107 seq(A130107(n), n=1..76); # _Peter Luschny_, Apr 06 2014
%t A130107 JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/#]&]/;(n>0)&&IntegerQ[n];
%t A130107 DedekindPsi[n_]:=JordanTotient[n,2]/EulerPhi[n];
%t A130107 A063659[n_]:=DivisorSum[n,MoebiusMu[n/#]*DedekindPsi[#]&];
%t A130107 A130107[n_]:=DivisorSum[n,MoebiusMu[n/#]*A063659[#]&]:
%t A130107 Table[A130107[n],{n,1,30}]
%t A130107 (* _Enrique Pérez Herrero_, Apr 03 2014 *)
%t A130107 a[ n_] := If[ n < 2, Boole[n == 1], Times @@ (Which[ #2 == 1, # - 1, #2 == 2, #^2 - # - 1, True, #^(#2 - 3) (#^2 - 1) (# - 1)] &) @@@ FactorInteger[n]]; (* _Michael Somos_, Jun 17 2015 *)
%o A130107 (PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k,]; if( e==1, p - 1, e==2, p^2 - p - 1, p^(e-3) * (p^2 - 1) * (p-1))))}; /* _Michael Somos_, Jun 17 2015 */
%Y A130107 Cf. A130107, A054525, A063659, A001615.
%K A130107 nonn,mult
%O A130107 1,3
%A A130107 _Gary W. Adamson_, May 07 2007
%E A130107 More terms from _Enrique Pérez Herrero_, Apr 03 2014