A130107 - OEIS

%I #35 Feb 20 2022 14:37:58

%S 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 8,30,12,20,16,24,5,36,18,24,12,40,12,42,10,20,22,46,12,41,19,32,12,

%U 52,16,40,18,36,28,58,8,60,30,30,24,48,20,66,16,44,24

%N Möbius transform of A063659.

%C Double inverse Möbius transform of A130107 = A001615: (1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, ...).

%H Enrique Pérez Herrero, <a href="/A130107/b130107.txt">Table of n, a(n) for n = 1..2000</a>

%H Steven R. Finch, <a href="/A001616/a001616.pdf">Primitive Cusp Forms</a>, April 27, 2009. [Cached copy, with permission of the author]

%F A054525 * A063659, (1, 2, 3, 3, 5, 6, 7, 6, 8, 10, ...).

%F 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 Dirichlet g.f.: zeta(s-1) / (zeta(s) * zeta(2s)). - _Álvar Ibeas_, Mar 07 2015

%F Sum_{k=1..n} a(k) ~ 270*n^2 / Pi^6. - _Vaclav Kotesovec_, Jan 11 2019

%e 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 with(numtheory): A130107 := proc(n) local dp, mtdp, d, p;

%p dp := n -> n*mul((1+1/p), p=factorset(n));

%p mtdp := n -> add(mobius(n/d)*dp(d), d=divisors(n));

%p add(mobius(n/d)*mtdp(d), d=divisors(n)) end:

%p seq(A130107(n), n=1..76); # _Peter Luschny_, Apr 06 2014

%t JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/#]&]/;(n>0)&&IntegerQ[n];

%t DedekindPsi[n_]:=JordanTotient[n,2]/EulerPhi[n];

%t A063659[n_]:=DivisorSum[n,MoebiusMu[n/#]*DedekindPsi[#]&];

%t A130107[n_]:=DivisorSum[n,MoebiusMu[n/#]*A063659[#]&]:

%t Table[A130107[n],{n,1,30}]

%t (* _Enrique Pérez Herrero_, Apr 03 2014 *)

%t 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 (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 Cf. A130107, A054525, A063659, A001615.

%K nonn,mult

%O 1,3

%A _Gary W. Adamson_, May 07 2007

%E More terms from _Enrique Pérez Herrero_, Apr 03 2014